Linear Differential Operators with Constant Coefficients

Table of contents :
Title
Preface
Table of Contents
Introduction
§ 1. Exponential representation for an ordinary equation with one unknown function
§ 2. Exponential representation of the solutions of partial differential equations
§ 3. The exponential representation of solutions of arbitrary systems
Part One: Analytic Methods
I. Homological Tools
§ 1. Families of topological modules
§ 2. The fundamental homology theorem
§ 3. Operations on modules
II. Division with Remainder in the Space of Power Series
§ 1. The space of power series
§ 2. The base sequence of matrices
§ 3. Stabilization of the base sequence
§ 4. p-decompositions
III. Cohomologies of Analytic Functions of Bounded Growth
§ 1. The space of holomorphic functions
§ 2. The operator Dz in spaces of type I
§ 3. M-cohomologies
§ 4. The theorem on the triviality of M-cohomologies
§ 5. Cohomologies connected with P-matrices
IV. The Fundamental Theorem
§ 1. Some properties of finite P-modules
§ 2. Local p-operators
§ 3. The fundamental inequality for the operator D
§ 4. Noetherian operators
§ 5. The fundamental theorem
Part Two: Differential Equations with Constant Coefficients
V. Linear Spaces and Distributions
§ 1. Limiting processes in families of linear spaces
§ 2. Functional spaces
§ 3. Fourier transform
VI. Homogeneous Systems of Equations
§ 4. The exponential representation of solutions of homogeneous systems of equations
§ 5. Hypoelliptic operators
§ 6. Uniqueness of solutions of the Cauchy problem
VII. Inhomogeneous Systems
§ 7. Solubility of inhomogeneous systems. M-convexity
§ 8. M-convexity in convex regions
§ 9. The connection between M-convexity and the properties of a sheaf of solutions of a homogeneous system
§ 10. The algebraic condition for M-convexity
§ 11. Geometrical conditions of M-convexity
§ 12. Operators of the form p(Dxi) in domains of holomorphy
VIII. Overdetermined Systems
§ 13. Concerning the modules Ext^i(M, P)
§ 14. The extension of solutions of homogeneous systems
§ 15. The influence of boundary values on the behavior of the solutions within a region
Notes
Bibliography
Subject Index
Index of Basic Notation

Citation preview

Die Grundlehrcn dcr mathematischcn Wissenschaftcn in Einzeldarstellungen Band I68

V. P. Palamodov

Linear Differential Operators with Constant Coefficients

Die Grundlehren der

mathematischen Wissenschaften in Einzeldarstellungen mit besonderer Berficksichtigung der Anwendungsgebiete

Band 168

Herausgegeben van J. L. Doob - A. Grothendieck - E. Heinz - F. Hirzebruch E. Hopf - H. Hopf - W. Maak - S. MacLane - W. Magnus M. M. Postnikov - F. K. Schmidt - D. S. Scott - K. Stein

Gescha'ftWhrende Herausgeber B. Eckmann und B. L. van der Waerden

V. P. Palamodov

Linear Differential Operators with Constant Coefficients

Translated by A. A. Brown

[in] Springer-Verlag New York - Heidelberg- Berlin 1970

Q3 ’1.“ “if

XII“; ‘3 ‘ Victor 1’. Palamodov

WW! II III: Univuflly of Malawi. U.S.S.R.

Arthur A. Brown Anon: Puk, Cambridle. W USA.

W Ember:

Prof. Dr. B. Eckmaun saw-cue 1mm 3mm 2mm. Prof. Dr. B. L. van der Wacrden ill-Wu 1mm def Univu‘iflt 2mm.

Translation of

Linejnye diflerencial'nye operatory s postajanuymi koqfficientami “Nauka”, Moscow, 1967

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Preface This book contains a systematic exposition of the facts relating to partial differential equations with constant coefficients. The study of systems of equations in general form occupies a central place. Together with the classical problems of the existence, the uniqueness, and the regularity of the solutions, we also consider the specific problems that arise in connection with overdetermined and underdetermined systems of equations: the extendability of the solutions into a wider region, the extendability of regularity, M—cohomology and so on. Great attention is paid to the connections and the parallels with the theory of functions of several complex variables. The choice of material was dictated by a number of considerations. Among all the facts relating to general systems of equations, the book contains none that relate to the behavior of differential operators in spaces of slowly growing functions. Missing also are results relating to a single equation in one unknown function: the correctness of the Cauchy problem, certain theorems on p-convexity, and the theory of boundary values, are all set forth in other monographs (Gel’fand and Silov [3], Homander [10] and Treves [4]). The book consists of two parts. In the first, we set forth the analytic method which forms the basis for the contents of the second part, which itself is dedicated to differential equations. The first part is preceded by an introduction in which the content and methods of Part I are described. All the notes and bibliographical references are collected together in a special section. This book was written at the suggestion of G. E. Silov. I am grateful to him for his unfailing support. I have also benefitted greatly by my constant contacts with V. V. Grusin. Let me make some remarks on how to use the book. For a first reading of § 1 of Chapter I, it is sufficient to limit oneself to the basic definitions. The rest of the content of this section is a detailed foundation for the argument contained in § 2. The content of § 3 of Chapter I, with the exception of the introductory section and 7°, is used only in the second part of the book, beginning with Chapter VII. Chapter Il—IV form an integrated whole. On a first reading of these, one may omit only §4 of Chapter IV, the content of which is used, in essence, only in Chapter VII. In §5 of Chapter IV only sub-

V'l

Preface

sections l°—3° need be read for an understanding of the remaining material. The general formulation of the fundamental theorem, contained in 4°——6", is used only in § 14 of Chapter VIII. In Chapter V, the content of §§ 1—3 represents, in essence, an exposition of more or less known facts from the theory of linear topological spaces, distributions, and Fourier transforms. In Chapter VI, 4° of §4, 8° of §5, and (56 are not connected with the subsequent material, and they may be omitted on a first reading. In the seventh chapter, §§ 11 and 12 are independent; § 13 of Chapter VIII is of auxiliary value only. In the first part of the book, the sections are numbered independently in each chapter. In the second part, the sections are numbered sequentially. Formulae are numbered according to the following rule: A formula with the number (a, b) is found in the section with the number b and has the serial number a. The citation of a formula relating to another chapter will contain within the parenthesis the number of the chapter cited, but the citation of a formula within the same section omits the section number. The symbol I] denotes the end of a proof.

Table of Contents Introduction ........................ (51. Exponential representation for an ordinary equation with one unknown function ................ §2. Exponential representation of the solutions of partial differential equations ................... §3. The exponential representation of solutions of arbitrary systems .......................

1

7

14

Part I. Analytic Methods

Chapter I. Homological Tools ................

17

§1. Families of topological modules ............ §2. The fundamental homology theorem .......... {73. Operations on modules ................

17 28 39

Chapter II. Division with Remainder in the Space of Power Series

52

§1. The space of power series ............... §2. The base sequence of matrices ............. §3. Stabilization of the base sequence ........... §4. p-decompositions ...................

53 58 68 75

Chapter III. Cohomologies of Analytic Functions of Bounded Growth .......................

88

§1. §2. §3. §4. §5.

The space of holomorphic functions .......... 88 The operator D5 in spaces of type J ........... 94 .ll-cohomologies ................... 105 The theorem on the triviality of .I-cohomologies ..... 110 Cohomologies connected with 9-matrioes ........ 121

Chapter IV. The Fundamental Theorem ............ 137 §1. Some properties of finite 9-modules .......... [37 §2. Local p-operators .................. 151 §3. The fundamental inequality for the operator 9 ...... 161 §4. Noetherian operators ................. 175 §5. The fundamental theorem ............... 186

Table of Contents

VIII

Part II. Diiferential Equations with Col-mt Coefficients Chapter V. Linear Spaces and Distributions .......... §1. Limiting processes in families of linear spaces ...... §2. Functional spaces .................. §3. Fourier transforms ..................

201 201 218 237

Chapter VI. Homogeneous Systems of Equations ........ §4. The exponential representation of solutions of homogeneous sytems of equations .................. §5. Hypoelliptic operators ................. §6. Uniqueness of solutions of the Cauchy problem .....

259

Chapter VII. Inhomogeneous Systems ............. §7. Solubility of inhomogeneous systems. Ill-convexity . . . . §8. M-convexity in eonvex regions ............. §9. The connection between M-convexity and the properties of a sheaf of solutions of a homogeneous system ....... §10. The algebraic conditions for M—convexity ........ §11. Geometrical conditions of M—convexity ......... §12. Operators of the form p(D3) in domains of holomorphy . . Chapter VIII. Overdetermined Systems ............ §13. Concerning the modules Ext‘ (M, .9) .......... §14. The extension of solutions of homogeneous systems . . . . §15. The influence of boundary values on the behavior of the solutions within a region ...............

259 268 283 289 289 299 316 335 346 356 370 371 386

410

Notes ........................... 423

Bibliography ........................ 432

Subject Index

....................... 441

Index of Basic Notation ................... 444

Introduction The last decade has seen the completion of the foundations for what today we call the general theory of partial differential operators with constant coefficients. The general theory is distinguished from the classical in that the study of particular properties of specific operators has given way to an investigation of the structural properties of operators of general form. Perhaps the best example is provided by the study of the local properties of the solutions of homogeneous equations. Special results on the regularity of the solutions of Laplace‘s equation, the heat equation and certain others laid the foundation for the singling out of classes of operators having similar properties: elliptic and parabolic. These classes of operators have a common property: Every solution of the corresponding homogeneous equation is infinitely differentiable. The next step was to pose the problem of writing down all differential operators having the same property. Such operators, which we call hypoelliptic, were completely described within the class of operators with constant coefficients. It was then observed that the regularity of solutions of hypoelliptic equations with constant coefficients is a simple consequence of a more general property which relates to all operators with constant coefficients. This general property consists in the possibility of an exponential representation and, more precisely, consists in the fact that every solution of the corresponding homogeneous equation can be written in the form of an integral with some measure over the set of exponential polynomials satisfying the same equation. Another line, which today characterizes the general theory, had as its starting point the classical problem of the construction of fundamental solutions in the large. Within the framework of the classical theory, this problem was solved only for certain special types of operators. Then, thanks to the application of distribution theory, the following general result was obtained: For an arbitrary non«zero operator with constant coefficients, there exists in the class of distributions a fundamental solution. Moreover, it turns out that the nonhomogeneous equation cor— responding to such an operator is soluble for an arbitrary right side. The subsequent stage of development along this line was connected with the consideration of a system ofequations (with constant coefficients)

Introduction

2

of general form. In this stage, the raw material for development of the general theory was provided by the corresponding portions of the theory of differential forms and functions of several complex variables. The result of their interaction was the general theorem on the solubility of an arbitrary system of equations with constant coefficients when the righthand side satisfies the “formal condition of compatibility." This theorem on the solubility of nonhomogeneous systems, together with the alreadymentioned theorem on exponential representations, provided, in its turn, a special case of the general theorem on exponential representations, which occupies an essential position in this monograph. A number of other problems in the general theory also lead to this general theorem. The second part of the monograph contains an exposition of the basic branches of the general theory of differential operators with constant coefficients, in the form of a series of consequences of the fundamental theorem an exponential representation. The proof of the theorem on exponential representation is the main content of the first part of the monograph and also of §4 of Chapter V. In order to help the reader, we now set forth in very brief form some of the simplest cases and some of the ideas of the proof. We presuppose that the reader is familiar with the fundamentals of the theory of distributions.

§ 1. Exponential representation for an ordinary equation

with one unknown function 1". Formulation of the problem. An arbitrary linear differential operator with constant coefficients in R' will be written in the form

p(D)= 2 ml)", where

.1" am D'=I —0€{' 66"“ .

pjeC.

lllén

. . . . . . . J=U;,...,j..)» |J|=J.+--~+J., l=V—1,

and §=(§1, ..., C") is some fixed system of coordinates in R'. Let 2: (2,, .. . , 2,) be a point of the n-dimensional complex space C". The polynomial , . 17(1)::s", z’=2i'--~Zi." iscalled the characteristic polynomial of the operator p(D). The algebraic variety N c C", formed by the roots of the polynomial p(z) is also called the characteristic variety. We choose some domain a: R' and consider the corresponding homogeneous equation

p(D)u=0,

(1.1)

§ 1. Exponential representation for an ordinary equation

3

in which u. is assumed to be a generalized function in (2. We note that for an arbitrary point z, belonging to the characteristic variety N, the function exp(z, —i é) satisfies Eq.(l). In fact,

p(D)exp(z, —i€)=p(2)exp(z, —i£)=0. On the other hand, if the function exp(z, — i 6) satisfies (1), then the point 2 belongs to N. We can also find other exponential polynomials satisfying (1). First of all, we take note of Leibnitz’s formula

piD)fg=zji!Di/pm(mg, 1'!=ii!---f.!, where pU’(z)=D£ 11(2). We apply it to the functions f=f(§) and g: exp (2, — i 6), where f(C) is a polynomial of order oi:

11(0)}(C) exmz, -i€)= MES D’/(€)P‘”(Z)€XP(2, —i§)-

(11)

It follows from this formula that if at the point zeN all the derivatives of the polynomial p up to order or vanish, then an arbitrary exponential polynomial of the form f(C) exp(z, —i.§) satisfies (1). We note that formula (2) implies also that for the exponential polynomialf(.5) exp(z, — i‘ 6) to satisfy (1), it is necessary that zeN 1. We can now formulate in approximate terms the problem of the exponential representation of the solutions of Eq.(1): To write an arbitrary solution of this equation in the form of an integral with some measure over the set of all exponential polynomials satisfying the same ’ equation. We now solve this problem in the most elementary case.

2°. The case of one independent variable. In this case Eq.(l) is an ordinary equation with constant coefficients, and the characteristic polynomial p(z) has only a finite number of roots {1, ...,C1. Let 111, ..., or, be the multiplicity of these roots; as is known, Za1=m, where m is the order of the polynomial p. From what we have said earlier, it follows that the exponential polynomial

C‘exm-iéii),

i=0,v--,ai—1, A=1,...,I,

(3.1)

satisfies (1) on the whole line. We note that an arbitrary exponential polynomial satisfying (1) is a linear combination of the functions (3). Thus, the problem of the exponential representation can be formulated now in these terms: To express an arbitrary solution of (1), defined in the open set (2, in the form of a linear combination of exponential 1 In Chapter V 57, we obtain a complete expression for the exponential polynomials satisfying (1).

Introduction

4

polynomials of the type (3) 2. Clearly, if this problem is to be soluble, the domain must be connected. Otherwise, we can construct a solution which is equal to distinct linear combinations of the functions (3) in the different connected components. In view of the fact that the functions (3) are linearly independent in an arbitrary open set, such a solution cannot be represented in the form of a linear combination of functions of the type (3) in the whole set 9. Therefore, we shall suppose that the region Q is connected, that is, it is represented as an interval (finite or infinite).

3°. Reduction of the problem. With no essential loss of generality, we shall suppose that the interval 9 has the form (—a, a) for some a>0. The symbol 9(9) will denote the space of all complex-valued infinitely differentiable functions on the line, of which the carriers belong to 9. By 9,, we shall denote for every b, 0 1, the operators dfi are constructed in a somewhat more complicated way; a case typifying the general situation is considered in Chapter IV, §4. Later we shall introduce an operator (1, which maps a function wezm) into the set of functions dflz, D) l/I(Z)IN‘. In order to describe the image of this operator, we generalize the notion of a function analytic on N. We define a holomotphic p—function3 as an arbitrary set {H}, of functions f[ defined on N‘, such that, for every point (e N, there exists a function F holomorphic in the neighborhood of C such that

fI(Z)=di(z, D) F(Z)|zu.

V1, 1.

The set of all holomorphic p-functions, having components that satisfy the inequalities

If1(Z)l§Cq(IZ|+1)" exp(y.(lm 1)),

q=0, 1, 2, ...; 3a.

is a linear space, which we denote by Z”((2). The operator :1 acts from the space Z (‘2) to Z, (f2) and vanishes on the subspace (p71, ..., E) Z (9). By the fundamental theorem of the first part of this book, this operator, in fact, sets up an isomorphism of the factor-space (2) with the space ZI, (a). From this result, by analogy with §2, we can derive an exponential representation for the solutions of the system (1)

u(§)=2 I die, — i E) exmz, -i6)ui-

(3-3)

1:; NA Here, for an arbitrary compact K,, the measures #51 can be so chosen that

E II(|Z|+1)‘°eXP(7.(ImZ))|IIiIax, and the sum and product of elements of A correspond to the

sum and the composition of these operators. We shall suppose that the module is unitary, that is, that the identity operator corresponds to the unit of the ring A. Since A contains the field C, every A-module

is at the same time a linear space over C. The module X will be said to be topological, if it is a topological linear space and if [or an arbitrary element aeA, the corresponding operator in X is continuous. Every A-module can be thought of as a topological module if we endow it with the discrete topology, that is, the topology in which every element is its own neighborhood.

18

l. Homological Tools

Let X be a topological A-module. Every submodule Y is a topological module in the topology induwd by X. We consider the factormodule X/Y, and the canonical mapping X —~X/Y, which carries every element of X into the coset to which it belongs. We use this mapping to introduce a topology into X/ Y, defining the open sets in the latter as the images of open sets in X. From now on, when we speak of submodules and factor-modules, we shall assume that their topologies have been introduced in the way just described. Let (1%: X —> Y be a mapping of A-modules. Several other modules are associated with the mapping: these are Ker ¢=kemel 43, Im¢= image 45, Coker¢= Y/lru ¢=ookernel 4), and Coirn ¢=X/Ker¢=ooimage 42. We shall suppose that the modules X and Y are topological. Then, by the argument just used, all four of the associated modules are also topological. We shall say that the mapping ()5 is continuous if it is continuous as a mapping of the topological linear space X into Y. In particular, the canonical mapping X —>X/Y is always continuous. From now on, all the modules that we encounter will be supposed

topological, and all mappings continuous (unless the contrary is explicitly stated). Therefore, the words “topological" and “continuous" will be omitted. Let ¢z X—>Y be a mapping of modules. and let X’, X":X’ and Y’, Y"c Y’ be submodules of X and Y such that ¢(X’)cY’. and ¢(X”)c Y”. Restricting ¢ to X', we obtain a mapping $: X’—> Y’. Since the mapping d} carries the submodule X” into Y", we can construct the

mapping

(5: X’/X”—> Y'/Y”

of the corresponding cosets. The mapping (5, is obviously a continuous mapping of topological modules. We shall call it the mapping associated with :15. The mapping (1%: X —->Y will be called an isomorphism and will be written X g Y, if there exists an inverse mapping 45“: Y—> X, that is, a mapping such that the compositions ¢¢“ and 115“ 4) represent identity mappings. We shall say that 45 is a homomorphism if the associated mapping (5: Coim (13» 1m (1) is an isomorphism. Clearly, every isomorphism is also a homomorphism. If the modules X and Y are given the discrete topology, then every mapping q): X —> Y is a homomorphism. A sequence of mappings

X —‘. Y —*» z is said to be semi-exact (at the term Y), it Im dz c Ker Ill. This sequence is said to be algebraically exact if 1m 45 = Ker up. Finally, we shall say that it is exact, if it is algebraically exact, and the mappings dz and up are homo-

§ 1. Families of topological modules

19

morphisms: In particular, the whole sequence is algebraically exact. if and only if it is exact when X, Y and Z have the discrete topology.

2°. Many-valued mappings of modules Definition 1. A many-valued mapping (1}: X —> Y of topological Amodules 18 a map from X into the set of subsets of Y, satisfying the following two conditions. It Linearity. For arbitrary x1, xzeX and a“ azeA, we have the inclusion relation a1 ¢(x1)+az¢(xz)c¢(al x1+az X).

(1-1)

11. Continuity. For an arbitrary neighborhood of zero VcY, we can find a neighborhood of zero UcX, such that for every e the intersection ¢(x)n V is not empty. It follows from Condition I that for arbitrary al, (115,4

“1 ¢(0)+412 ¢(0) X, which carries every coset into the set of elements that belong to it. We shall call this a canonical mapping. Every ordinary, i.e.. singlevalued, mapping is also many-valued. From now on, unless the contrary is explicitly stated, we shall assume that all mappings are many-valued.

20

I. Homological Tools

Let d): X —> Y and l/I: Y—> Z be two module-mappings. We define their composition. For every xsX, we denote by 1/1 ¢(x) the union w(¢(x)) of the sets (My) for ye¢(x). Let us prove that the mapping ill 4: is linear and continuous. Using the linearity of the mappings 45 and W we obtain

at V’(¢(xi))+ a2 '/’(¢(xz)) C Mai ¢(Xi) +42 ‘Nxzn C ¢(¢(“r x1 + “2 x2»

for arbitrary xl, xzeX and a“ azeA. This verifies the linearity. Let W be a neighborhood of zero in Z. Since 4!) and l]! are continuous, we can find neighborhoods of zero Vc Y and UcX such that for an arbitrary xe U the intersection ¢(x)n V is not empty, and for arbitrary e the intersection w(y)n W is not empty. It follows that for arbitrary x e U, the intersection :11 (qb (x)) n W is not empty. We have thus established that the composition 1/145 is a mapping. 3“. Families of topological modules Definition 2. A family of topological A-modules will mean an arbitrary system X = (XI, if), consisting of the functions X“, defined on the set of all integers with values which are topological A-modules, and of the set of single-valued continuous mappings if: X, —>X,., defined for arbitrary pairs of integers at and at', and such that uga'. and satisfying the following conditions: a) for arbitrary a the mapping i; is the identity, and b) for arbitrary an; ar’ go”, we have 3:7 12' = 1:". In some applications, we shall deal with families {X,, if}. in which the modules XI and the mappings if are defined only for 0 < ué tx’. Such incomplete families can be completed in a trivial way by setting X,=0

and i:'=0 for go. Definition 3. Let X=(X,, if} and Y=(Y.,j:'} be two families. A family—to-family mapping 4): X —vY is defined as an ensemble of continuous mappings

15.1 X.-* YM, which are defined for all integer a. where 3(a) is a nondecreasing function of a tending to too together with a. and for arbitrary oz Y is said to be an identity mapping if its components are the mappings if“). Thus, the identity mappings can be put into one-to-one correspondence with their orders, that is, with the non-

decreasing functions Ma) which tend to 1 00 together with a. If (“00501, the corresponding identity mapping is said to be a unit mapping, Let X: {X,1‘;'} be a family The family Y: {Y ,f;'}'15 said to be a subfamily of the family X if for every a, Y is a submodule of X (its topology being induced by the topology of XI), and if the mappings 1‘. are restrictions of the mappings i: .Let Y be a subfamily of X. We shall consider the sequence of factor-modules X,/ Y, Since for arbitrary a’ > a,

1‘:’(Y,) =j:'(Y)c Y we can define the mapping 11': Xa/Y —>X /Y, associated with the mappingi1‘ It is easy to see that the modules Xa and the mappings i form a family. This family we shall call the factorfamily and denote it by X/Y. The canonical mappings XflaXfl/YIL and Xw/Yfl—tXnI define mappings of the families XaX/Y and X/Y—>Y, which we shall also call canonical mappings. Let X:{Xfl, if) and Y:{Y,,f;'} be arbitrary families and let 4): ((1),: X, —> Yam} be a mapping of them. Since the diagram (2) is commutative, the mapping if carries the submodule Ker 35, into the submodule Ker d3". Therefore the submodules Ker 45‘, with the topology induced by the modules XI, form a subfamily of the family X. We denote this subfamily by Ker 4:. Now we construct the image of (15. For every integer [3 we consider in the module Y the submodule jihad), (X ), where a is the largest > Mac). If we assign to these submodules the topology number such that [3: induced by the topology of the modules Y,, we obtain a subfamily of we denote by Im (15 or ¢(X). We shall consider which Y, the family also the factor-families Coker ¢= Y/Im (p. Coim ¢=X/Ker d), Let

X—»" YL z,

¢=f¢ii IKE—Him},

1/I={III.= Ya-tlml

be a sequence of famllies and mappings. The composition of the mappings will be the mapping with components Illam” Y be a mapping and I: XHX be an identity mapping. Then we can find an identity mapping .1 of the family Y such that d) I = J ¢. Proof. Let us establish the first assertion. Let 3‘01) be the order of the mapping 1,, i = 1,2. We write 13(01):m film). Now we extend both of the nondecreasing functions yi: [Mada [3(a) to all integer values of the argument while preserving the property of monotone growth. We take the functions y,(a) as the orders of the mappings Ii+ 2 , i = l, 2. Let us prove Part 11. Let [3(a) and y(a) be the orders of the mappings (15 and J. We choose the function y(fl(a)), which is the order of J ‘15. to be the order of the mapping 1, and the function [3(a) as the order of J’. Let us prove Part 111. Let 6(a) be the order of the mapping 1. We extend the function fi(a)—> fi(6(ot)) to all integer values of the argument, preserving the property of monotone growth, and we take it as the order of the mapping J. I] 4°. Equivalent families and mappings Definition 3a. Let X' and X” be subfamilies of some family X. We shall say that the subfamily X’ rnajorizes the subfamily X”. and we write X’>X”, if there exists an identity mapping I of the family X, which carries the subfamily X” into X’. Let X’, X”, X"' be three subfamilies of X, satisfying the relations

X’>-X">X"', that is, let there exist two identity mappings I' and I", which respectively carry X” into X’ and X"’ into X”. Then the composition I’ I” carries X”’ into X' and therefore X’>X"'. Thus, majorization is an ordering relation. Let X’ and X”-X2, if we have simultaneously X1>X'2 and X;’>X’1’.

§ 1. Families of topological modules

23

Majorization, as applied to families associated with X, is an ordering relation also, since the majorization of subfamilies is an ordering relation. If we make use of Proposition 1, Definition 3 b can be rewritten as follows: X1>X2 if and only if there exists an identity mapping I: X —>X. such

that I(X’2)c X; and 1(X;)c X;'.

We shall say that the families X 1 and Xz are equivalent, and we write X1~X2, if we have simultaneously X1>X2 and X1X“, the factor-family X’/I (X") is independent (up to an equivalent) of the choice of the identity mapping I which carries X” into X’. For this factor-family we shall employ an abbreviated notation X’/X”, which does not mean, however, that X" is a sub-family of X’.

Definition 4. We shall say that two mappings dz, 41': X ., Y are equivalent, and we write ¢~¢', if there exist identity mappings J, J': Y—a Y, such that J 45 =J’ 41'. Let us validate this definition. Suppose that 45 ~ qb’ and 41’ ~ 4)“, that is,

J¢=J' 41',

1* 41:1" 41".

According to Proposition 1 there exist identity mappings J1 and J2 such that J1 J’ = J2 .1“. Hence, J!) 11) =J2 J“ 115", that is, ¢~¢". Proposition 2. The equivalence relation is preserved under composition. Proof Let ¢, (11’: X .4 Y and 1/1, 111’: Y—>Z be pairs of equivalent mappings. Then by definition J ¢=J'¢’ and K 1//=K’ up’ where J, J’, K, and K' are identity mappings. By Proposition 1, we can find identity » mappings Kl, Ki: such that

K,K¢=K¢J, Then

K1K'¢'=K/.n

K1 KIII¢=K¢J¢=K'III'J'¢'=K1 K’vli’tfi',

i.e.,¢¢~1/I’¢’. I] In particular. for arbitrary mappings 4): X —> Y and identity mappings I: X —>X, the mappings «111 and 4) are equivalent Now suppose there 1s given a diagram of mappings of families, that is, some set of families X‘, 1'6], and the set of mappings ¢{: X'—>X’ defined for some pairs (1, fie} x}. We shall say that the diagram 1s commutative if for any pair (@351 x J, all the compositions of the form

1:1. ...¢-::’ ¢-": X‘—»X’ are equivalent. Propositionz implies that the diagrams remain commutative when all the mappings are replawd by equivalent mappings. Definition 5. Let X’ and X" Y’,

gs": X“—> Y"

be mappings of these families. Then, we shall say that the mapping 45’ majorizes the mapping 4)", and we write ¢'>¢", if there exist identity mappings l and J of the families X and Y, such that the mappings I and .7 associated with them define the commutative diagram: Xti, Y’ I J

(3.1)

XML. Y” that is, ¢’7~.7 (15". It is obvious that the majorization of mappings is an ordering relation. If X’~X”, Y'~Y", ¢'>¢", and ¢”>¢‘, then we shall say that the mappings 4)’ and ¢” are equivalent and we write ¢'~¢". If X’=X” and Y‘= Y", the equivalence relationship just defined coincides with the equivalence relation set up in Definition 4. In fact let (15’ and d)” be mappings that are equivalent in the sense of Definition 5. Then there exist identity mappings J1, J2: Y'—v Y', such that J! ¢’i=J,]¢”. By Proposition 1 we can find an identity mapping J‘ such that J' Jl ¢'=

J1 ¢’I. Thus, J* J1 45’ =12 .7 4)”, that is, 42’ and 4:" are mappings equivalent in the sense of Definition 4. Let X’, X” be families associated with X, and let Y', Y” be families associated with Y. We shall suppose that X”X’ and J: Y’—> Y”, associated with identity ma pings. In this case, for any mapping 45: X ’—> Y’, the compositions ¢ : X”aY' and I45: X’—*Y” are defined. It is clear that these compositions (up to an equivalent) do not depend on I and J. From now on we shall make no distinction between the mappings $7 and

Id) and the mapping (15 itself. Now suppose X and Y are arbitrary families and X’/X" and Y'/Y” are families associated with them. Suppose further that we are given the mapping (1%: X HY, satisfying the conditions ¢(X')< Y' and ¢(X“)< Y”. Replacing dz by an equivalent mapping (if we have to) we have the inclusion relations ¢(X’)c Y’ and ¢(X")c Y". Then we consider the mapping (52 X'/ "a Y’/Y",

which has the components (5‘: X;/X;'a Y,’/)1,” associated with the components of the mapping (1). We shall call c} the mapping associated to 43. It is easy to see that the construction we have just gone through

defines :5 up to an equivalence. Let 43’, ¢”: X —> Y be two family-mappings having the same orders. By the sum of two such mappings, we shall mean the mapping ¢’+¢":

§ 1. Families of topological modules

25

X —> Yhaving as components the sums of the corresponding components of dJ’ and (1)”. If ¢’ and 43” have different orders, their sum is the mapping J’ ¢’+J”¢”, where the identity mappings J’ and I” of Y are so chosen that the mappings J’ 45' and J”¢” have the same orders. It is clear that the mapping ¢’ + 45" (up to an equivalent) does not depend on the choice of J’ and J". It is clear that the addition of family mappings and the multiplication of a mapping by an element of the ring A does not destroy the majorization relation introduced in Definition 5. Let X' and Y’ be associated to X and Y, and let 4): X’—> Y' be a

mapping of these families Then the families Kerdz and Coim¢ are associated with X' and, accordingly, with X also; the families Im¢ and Coker¢ are associated with Y’ and, therefore, also with Y.

Proposition 3. Let the mappings ¢’: X’—> Y’ and 43": X"—>Y" satisfy the relations X’>X", Y’>Y”, ¢’>¢”. Then we have the inequalities

Irn ¢’>Im ¢",

Coker ¢'>Coker 43”,

Ker ¢’>Ker 4:”,

Coim ¢’>Coim ¢”.

4l (

)

Proof. Since the diagram (3) is commutative. there exist identity

mappings J1 and J; of the family Y’ such that Jl¢'i=.]2 Ida”. Since the mapping J2] carries [m (15” into Im sd)”, the family ImJIIqS” majorizes Im d)”. On the other hand,

Isi¢"=ImJ,¢'ic1m¢'Ic1m¢'. Accordingly, the first of the relations (4) is proved. The second follows from the first, since Y'> Y". Let us now prove the third relation. Propositionl implies that there exist identity mappings Io, J3 of the families X" and Y’ such that

¢'i10=J,J,¢'I=J,JZJ¢". Since the mapping ill, carries Ker 4:710 into Ker¢', we have Ker w c Ker J, J2 J 4:" = Ker 4H IoZ".

¢'>¢"‘

'l”>ll’"~

26

l. Homological Tools

Then ¢’¢’>¢"¢”. In other words, our composition preserves the order relation of mappings. Proof It is easy to see that the commutativity of the diagram (3) is independent of the choice of the mappingsI and J. Hence. for arbitrary mappings I: X”—»X'. .7: Y”—» Y' and K: Z"—-> 2', associated with identity mappings in the diagram

x’—»" Y’ —»*’ 2'

ii

i

i:

X" 4," Y" 0" Z" both squares are commutative. Accordingly, by Proposition 2 the entire diagram is commutative. Therefore I//’¢’>|]/"¢". I] Definition 6. The mapping 4): X—> Y is said to be a monomorphism if Ker¢~0, and an epimorphism if Coker¢~0.

By Proposition 3, this definition is invariant when the mapping d: is replaced by an equivalent one. Proposition 5. If in the sequence

W—»" X—>‘ Y——»‘ z the mapping 4) is a monomorphism, then Ker¢ x~Ker 1. If (It is an epimorphism, then Im d1 ¢~Im w. Proof Let 4) be a monomorphism. Then, since Kemp ~0, we can find an identity mapping I of the family X such that I (Ker ¢)=0. Now, let weKer ¢ 1, that is, 4) 1(w)=¢ x(0). This equation implies that we can find an element xex(w) such that ¢(x)= ¢(0), that is, xeKer ¢. This means that I(x)=0, hence, the set I 1(w) contains the zero element, and therefore coincides with I 1(0). Thus, Ker 4) cerI 1. Proposition 3 implies that Ker I x~Ker x, whence Ker ¢ x Y establishes an isomorphism, if there exists a mapping 4)": Y—>X such that

¢¢"=J,

¢"¢=I.

where I and J are identity mappings of the families X and Y. We write: 0 X g Y.

§ 1. Families of topological modules

27

. The equation 4) ¢“=J, implies, in particular, that the mapping o [5 equivalent to a single-valued mapping. We shall refer to ¢_l as the inverse of d). We shall say that the mapping (l: is a homomorphism, if the associated mapping ‘ d): Coim if) —v [m ()5 is an isomorphism. Thus, every homomorphism is equivalent to a single-valued mapping. If the mapping (1: is an isomorphism (homomorphism), then every mapping equivalent to it is also an isomorphism (homomorphism).

Proposition 6. The mapping 95: X a Y is a homomorphism if and only if there exists a mapping d)’: [m ¢~>X/Ker¢ such that the composition 41 (19’ is equal to an identity mapping of the family lm 43.

Proof. Necessity. It follows from the definition that there exists a mapping ¢—*: Im d) a X/Ker¢ such that the composition 6 d)“ is an identity mapping. Therefore the composition of db“ and the canonical mapping X/Ker¢ 4X forms the desired mapping (1". Sufliciency. We denote by at ‘ ‘ the composition of 45' and the canonical

mapping X—>X/Ker¢. Since $¢“=¢¢’, the composition «545—1 is an identity mapping. Thus $¢—1¢~$, that is, $(¢“$—j)~0 where j is a unit mapping of the family Coim 4). Since Proposition 5 implies that

is a monomorphism,

Ker(¢" $-J)~Ker$(¢"$-J)~C0im ¢, whence ¢“J—j~0. I] Definition 8. Let X 4» Y—"> z

(51)

be a sequence of families and mappings which are equivalent to singlevalued mappings. We shall say that this sequence is semi—exact or it is a complex if Im ¢0 be an integer. By [X]" or X" we shall denote the direct sum of K summands which are each equal to X. For the sake of generality, we shall write [X]°=0. If 45: X —>Y is a module mapping, then [¢]" will denote the mapping

[X]"9(Xi, xii—4450C), ¢(Xt))€[Y]*If X ={X,,i‘,‘_’) is a family of modules, then [X]" will denote the family ([Xu '5 [5],? Let XAY—hz (6.1) be a sequence of families and mappings, and let

[X]*L'”»[Y1*L"»EZJ*

(7.1)

be a sequence of powers of these families and mappings. Then, if d) is a homomorphism, it is clear that [¢]“ is also a homomorphism If the sequence (6) is exact, then the sequence (7) is also exact, and so on. In order to avoid notational clumsiness, we shall often write :1) and 1/1 in sequences of the type (7) in place of [(15]“ and [111]“. The total effect of Sections 3°-— 6° is that we may neglect distinctions between equivalent families and between equivalent mappings. We note that any topological module X can be considered as a family if we set X,2X and we choose for the mappings if the identity mapping of X. If X and Y are two modules and 4): X —> Y is a mapping between them, then 4) can be considered as a mapping of the corresponding family In this sense, the definitions given in 3°—6° as applied to families and mappings of this type coincide with the corresponding definitions given in I“. § 2. The fundamental homology theorem 1°. Theorem 1. We consider three types of commutative diagrams (as shown immediately below) formed offamilies X{ of topological modules and single-valued mappings of them.

§2. The fundamental homology theorem

29

I. Let us suppose that all columns and all rows of distinguished parts ofthese diagrams are exact. Ifone ofthe mappingsfor g is a homomorphism, the other is a homomorphism also. II. Let us note the extreme rows and columns marked with the symbols f, f‘, g, and g’, and suppose, in addition to the condition stated in the first part of the theorem, that all rows and columns of our diagrams, with the exceptions of those already noted, are algebraically exact. Then the extreme rows and columns are semi-exact and we have the natural isomorphisms (or)

0—-—>X:

——>X,f

I. Homological Tools

30

Ker f’/Im f cr g’lIm g.

(1.2)

We postpone the proof for 2". Three lemmas Lemma 1. Let us consider the commutative diagram

Afiafia h] in

(2.2)

Yz —‘» Y3 0 of mappings offamilies and suppose that the right-hand column is exact, that the first row is algebraically exact, and the mapping: 4) and W1 are homomorphism. Then, A) The mapping pg: z1—»z,/1m 1,, which is an extension of the mapping 1111 (that is, it is a composition of W1 with the canonical mapping Z2 —> Zz/Im 1,), is a homomorphism; B) If Zl ~ 0, then the mapping x; is also a homomorphism. Proof. Let

KIImW+NWr”m%+mWflmh=m¢b x": 1m wg—vlm ¢1+Im 12

§2. The fundamental homology theorem

31

be canonical mappings. Since the first column of (2) is semi—exact, it follows that the restriction of the mapping '1’; on lm llll + Im x2, in composition with some identity mapping, acts to Im :11; Xz- Because the diagram (2) is commutative, Im V12 12~lm X3 ¢, whence I1/1(lm ¢1+ Im Kz)< Im x, 47; therefore, the mapping 1/12 (in composition with some identity mapping) carries 1m l/ll-l—Il'n 12 into [m x; 4).

Because the right-hand column is exact. there exists a single-valued mapping x; 1: Im x3 —> Y3 , which is inverse to 13 . We consider its restriction on the subfamily Im 13 4). Since 13" 13 ¢~¢, the restriction carries Irn x34) into the subfamily Im¢ of the family Y. Since ()5 is a homomorphism, the inverse mapping (1)“: 1m (1) —> Y2 is defined. We consider the composition m=12¢"1§‘ W2, which acts from Im ¢1+Im x; to Im 12. Since $2 11~13 q), we have '1’: W~X3¢¢_1X§1'/’2~'/’1So, for the mapping 6: (i —a)) 16—1. which acts from [In 1111 to Z; , we have up; 5~0, that is, Im 6Z,. It is clear that

ll/illl{15~K6=j—Kwk_l

where j is a unit mapping of the family Im l/Ii. Since the mapping to acts into Im 12 , the composition Kw is a null mapping. Therefore, {#1 :11; 1 6 ~j, from which (by Proposition 6 of§ 1) it follows that $1 is a homomorphism. This proves assertion A). We now prove assertion B). Suppose that Z1 ~0. Since the right-hand column of (2) is exact, the mapping z, is an isomorphism from Im :17 onto Im 13 ¢ ~ 1m III, 12 . Therefore, we may consider the composition 11 Imxz—‘h—Jm Wzlzfi’lm‘i’os—fl’ Y2-

Furthermore, we have

III; 12 X='/’z z; ¢" 13“ lllz~x3 M“ x3“ tam/12, that is, l/Iz(11 x— i)~0, where i is a unit mapping of Z1. Since the first row is exact, Ker 1/1, ~0, and therefore, X2 x~i. It follows that x; is a homomorphism. I]

l. Homological Tools

32

:11: N-—>

T—r_rf—> o

Lemma 2. Let us consider the commutative diagram

x2 :23

of mappings of families and suppose that the leftt column is exact, that the lowest row is algebraically exact, and that 4) and 1/12 are homomorphisms. Then A) The mapping 1V2: Ker 12—»23, a restriction of [1/2, is a homomorphism;

B) If Z3 ~ 0, then X; is also a homomorphism. Proof It follows from the conditions that there exists a mapping up; 1: Im 1/12 fizz/Ker [[11, inverse to 1/11. This mapping carries the subfamily Im 1V, into the image of the canonical mapping Ker 12—» 22/ Ker $1, that is, into the family (Ker 12+Ker upg/Ker I112. Let A: 21/ Ker ill, —> Z; be a canonical mapping The composition Ail/2‘1 carries Itn W2 into the sum Ker 11+ Ker l/I,~Ker x; +Im $1. Let us now consider the sequence of mappings

—‘+Im zzwt~lm¢x1~lm¢‘—"> y, —">z Kerx2+1m uh"

' Ker w,

The composition of these mappings will be denoted by p. Since 12 ¢1~ (151:, we have K2 P~¢Ki 11—1 ¢_llz~Xz~ Thus Xz(i—P)"0, 811d. therefore, the mapping (i— min/12“ (up to an equivalence) acts from Im 1/"; to Ker 12. Moreover

I/I'z(i-P)l¢2"='/I21¢I2“~k, where k is a unit mapping of Z3. It follows that Il/2 is a homomorphism, and with this, assertion A) is proved. We prove assertion B). Let Z3~0. Then. since the lefthand column and the lower row are algebraically exact. we have Irn 12~Im d). We consider the composition of the mappings 1’:- Im 12 —’ Yil—’ _’zl—’ Zr The mapping x’z satisfies the relations

x213=12¢111¢d~¢1111l¢—‘~ where l is a unit mapping of Y2; therefore, x, is a homomorphism.

I]

9'2. The fundamental homology theorem

3]

n

in;—» 1*N

(3.2)

o —'~’
all—$21. VIII/“=Illxl ¢" x;'~12 ¢¢“xi‘~i,

where i is a unit mapping of Im :11. We now prove assertion b). Since I: is an epimorphism and the diagram (3) is commutative, we have the equivalence [m 11 ¢~ lm 1/1 1, ~ Im 1/1. The desired mapping 4)" is the composition of the mappings

Im¢—’l>Imt/I‘—_'>ZI—£>Yl. Since 12 is a monomorphism, the relations

12 ¢¢“=xz M” II!" Zz~'/’11 xr‘ III’l xz~xz imply that (15:15" ~j, where j is a unit mapping of Y2. This proves asser— tion b). [I 3°. Proof of the first assertion of the theorem. We consider the dia— gram (0:). Let us suppose that the mapping f is a homomorphism. We then show that the mapping g is also a homomorphism. In the case n = 2, the proof follows from Assertion B) of Lemma 1. Supposing that n>2, we apply Lemmal to the portion of the diagram (oi) formed by the two lowest rows and the columns with the numbers n—-3, n—2. and n— 1. This shows that the mapping

Xz-n-Z/Ii-Z, tremor-Zora,

(4.2)

34

l. Homologieal Tools

which is an extension of the mapping X‘2'" —> X5‘ 1, is a homomorphism. We note that the family 1;”: belongs to the kernel of the mapping Xg- ’ —>Xg‘ 2. Thus, we may form the associated mapping

Kym-MK?! We now consider the diagram: X',‘“—sX;'3 __,X;—z

X;*°—»X;-'/r:-‘ 0 and we again apply Lemma 1. We note that all of the mappings involved are homomorphisms, and that the rows and columns are exact. Therefore the mapping

X;“—»X;~’/I:".

which is defined in the same way as (4), is also a homomorphism. A repeated application of this argument brings us to the diagram: 0 —’ XLi —’ XL 1 l

XL: —’X3—1 12—2

0 in which the rows and columns are exact, and all the mappings, with the possible exclusion of g, are homomorphisms. If we apply assertion B) of Lemma 1, we see that g is also a homomorphism. This proves the first assertion of the theorem for the diagram (at). We now take up diagram ()3). Suppose it is known that f is a homomorphism. A repeated application of Lemma 1 leads us. as in the preceding case, to the diagram

0

x:——Hx: X:-l—’X2-l/Iz—1 0

§ 2, The fundamental homology theorem

35

in which both columns are exact, and all the mapping, with the possible exception of g, are homomorphisms. If we apply the assertion a) of Lemma 3, we see that g is also a homomorphism. Let us now suppose that g is a homomorphism. Applying Lemma 2 to the portion of the diagram ([3) formed by the first two rows and the first three columns, we conclude that the mapping

Ki—IHXE—l’

K:,1=Ker(Xf_1—v),

(5-2)

which is a restriction of the mapping X}, 1 —>X,,3_l. is a homomorphism. We note that the image of the mapping X}_14X}_1 belongs to KLI. We may therefore consider the diagram: 0 Kz—lfi'ail X:—2 HXS—z —’X:-z

All the mappings in this diagram are homomorphisms, and the rows and column are exact. Applying Lemma 2, we find that the mapping 3 4 Kll-z “xii-1s which is an analog of (5), is also a homomorphism. By repeated applications of this argument, we are led to the diagram: 0

Kr‘aX; Kim—“X: 0

in which both columns are exact, and all the mappings, with the exception, perhaps, of f, are homomorphisms. But assertion b) of Lemma 3 implies that the mapping f is also a homomorphism. We have thus proved the first assertion of the theorem for diagram (1}). Let us now prove this assertion for diagram (y). In order to obtain the implication “g is a homomorphism implies f is a homomorphism," it is sufficient to make a repeated application of assertion A) of Lemma 2 (as we did before), and then to apply assertion B) of the same lemma. The converse implication reduces to what we have proved by the sym-

36

l. Homologieal Tools

metry of diagram (7) with respect to its bisectors. Thus, the first assertion of the theorem is completely proved. 4". Proof of the second assertion of the theorem. We shall first prove a lemma.

Lemma 4. Suppose that in the diagram

2 i» z’

I. y'i. Y x is a homomorphism and W x ¢~0. Then there exists an isomorphism [Im aer lid/1m 1¢‘:—:Ker¢x/[Kerz+lm 4)],

associated with 1. Proof. Since 1 is a homomorphism, there exists an inverse mapping 1“: Im 1—» Y/Ker 1. The restriction of this mapping on [m X n Ker III carries the subfamily into the family Ker n11 x/Ker x. The mapping 1“ establishes an isomorphism between the subfamily Im 1 45 c Im x n Ker I]; and the image of the mapping 1m ¢—> Y/Ker x. This image is clearly equal to the family [Ker x + Im ¢]/Ker 1. Accordingly, the mapping associated with X establishes an isomorphism [Im 1 n Ker fl/Im x ¢§(Ker w x/Ker x)/([Ker x + Lrn ¢]/Ker 1) gKer uhx/[Ker 1+ [in 43].

[I

Let us now turn to the diagrams (a), (If), (y) and prove that the extreme rows and columns are semi—exact. We show, for example that f’f~0 in the diagram (at). The composition of the mappings f’f and the monomorphism XfAX; is equal to the composition of the mappings

Xf’zaxg‘ZAXg" —»X;: the composition of the two latter mappings is zero. Hence f’ f~ 0, We now establish the isomorphism (1) for the diagram (at). For every pair (i, j), satisfying the relationship i+j=n—l or u, there is defined a mapping X{—>X{Il‘, as the composition of the mappings

XiaXi+1~Xm or Xiaxai #2:;

(6.2)

These compositions are equivalent since the diagram is commutative. For any pair (1‘, j) with i+j=n or n+1, we consider the family H{, associated with X{, which is defined for i +j = n by the formula

H{=Ker(X{—> X;';,‘)/[Im(X{‘l —>X{)+Im(X{_l —> Xh],

§ 2. The fundamental homology theorem

37

and for i+j = n +1 by the formula

H{ = [Ker (X! —» xg“) n Ker(X{ —> X{+,)]/Im(X{_‘l‘ —> X{). We now apply Lemma4 to the fragment:

Xi“

. —»X,.i _.x{+1 i-1

Xi—l

,

J'

Xiil

,

i

i+j=.n+l. It can be easily shown that the application of this lemma yields the isomorphism .

Inmate,

(72)

associated with the mapping X{_1—>X{. In the same way, we can establish the isomorphism

H{' ‘ gag,

(3.2)

associated with the mapping X!“ _. XI. Combining the isomorphisms (7) and (8), we are led to the isomorphism H:_I;H:“. We note that

H:_1~Kerg’/Im g,

H',‘"~Kerf’/Imf,

and obtain (1). The construction of the isomorphism (1) for the diagrams (fl) and (y) is carried out in the same way. I] Corollary. Suppose that in the diagrams (or), (,8), and (y) all rows and columns are exact, with the exception of one extreme row or column. Then this row {or column) is also exact. 5°. Refinement. I. For the sake of simplicity, we shall suppose that the diagrams (a), (fl), and (y) are strongly commutative, that is, for arbitrary i and j, both of the through mappings (6) coincide (they are not only equivalent as has been assumed up to now). We shall also suppose that all the rows and columns are strictly semi-exact, that is, the composition of two mappings in any rows or columns is equal to zero. We note that in the applications of Theorem 1, these conditions will be fulfilled in almost all cases.

l. Homological Tools

38

The hypotheses of the second part of the theorem amount to the statement that there exist (generally speaking, many-valued) mappings Ker(X{+‘—+X{”)—>X{,

(9 2)

Ker(X{H-’Xi+z)-»Xl,

'

which are inverse to the mappings

X{—>X{+',

X{—»X{+,.

(10-2)

0n the other hand, the assertion of the second part of the theorem

consists in the statement that there exist mappings Ker f'/Im f —:>._Ker g’/Im g, which set up an isomorphism. If we inspect the proof of the theorem, we can see that the mappings I and I can be effectively constructed, and are polynomials in the mappings (9) and (10). Therefore, the orders (in the sense of 3" § 1) of the mappings I and J are well-defined compositions of the orders of the mappings (9) and (10) and, therefore, depend only on the orders of these mappings, and not on the mappings (9) and (10) themselves 11. The following result shows how far we can weaken the hypotheses of Theorem 1, and still have it continue to act in one direction (in this case, from above downward). Theorem 1’. Suppose that in the commutative diagram (at) all rows and columns are semi-exact and that the sequences a

”d

Xl—rxr‘axrz. X{_,—rX{—»X{+,,

i+j=n i+j=n

i=2.-...n (X2=0), i=l,...,n—l (X'=0),

are algebraically exact. Then the bottom row of the diagram is also algebraically exact. The proof of this theorem is left to the reader. 6°. Mappings of diagrams of types (a), ([1), (y) Theorem 2. Suppose given two diagrams of type (at) having the same size

(Xl;flf’.g.g'),

{%i;/./.y.y'}.

(11.2)

and satiajying the condition of Theorem 1. We suppose further that for an arbitrary pair (i,j) with i+j=n—1, n, n+1, n+2, there are mappings ¢{: X{—>X{, such that the mappings a}! commute in the ensemble with the mappings of the diagram (11). Then the associated mappings L1 and

53‘ Operations on modules

39

(52“ form a commutative diagram:

Ker/’/[mxé Ker 971m , Jeni

Ia».

(12.2)

Kerf/lmfé Ker g’flm g in which I and J are isomorphisms of type (1). A similar assertion holds for diagrams of type ([3) and (7). Proof. We shall consider only diagrams of type (on). It follows from our hypotheses that mappings associated with ¢{ carry kernel and image of the mapping X{—>X{“ respectively into kemel and image of the mapping exam!“ and so on. Therefore, for an arbitrary pair (i, j) with i +j =n, n+1, we may define the mapping H{ —> If, associated with ¢{ where 916." is a family analogous to H{, constructed for the diagram (X{}. Then, we have the commutative diagram: ”Einzxé’sxi’"

i+j=n+1,

H{_,; H{;H{“ in which the rows contain isomorphisms of type (7) and (8). Combining these diagrams, we arrive at the commutative diagram

”TIE-”$.51 Hf“;H;_l which coincides with (12).

[I

§ 3. Operations on modules In this section, we introduce the functors ® and Ham, and their deriv-

atives1 and we establish certain properties of these funetors, limiting ourselves to the case in which the ring is a Noether ring and the first argument is a finite module The facts which we set forth here are in general well known; they can be found for example in the text of Zarisky and Samuels [1] and Cartan and Eilenberg [1]. For the convenience of the reader, however, we supply the proofs of some of them. 1 We assume that the reader has at least a superficial acquaintance with the concept of functor; see, for example, Godemerrt [l] §l. Chapter I.

I. Homologiml Tools

40

The symbol A will again be used for an arbitrary commutative ring over the field C. All the A-modules that we shall encounter will be assumed to be topological, and all A-mappings are assumed to be continuous and single-valued. Let 45 be an A-module and let k be an integer. The module 45“ will be interpreted as a module formed of columns of length k, the components of which belong to d}. The module A" has a canonical basis, formed of columns 2,, i=1, ...,k, of unit matrices of order k.

Let p: A‘—vA’ be an A-mapping. Such a mapping is characterized by its values p(e,) on the basis elements of the modules A'. We denote by p a matrix of size txs, formed of the columns p(e,~), i=1, ..., s. It is clear that the mapping p is effectively a multiplication of the columns of A“ on the left by the matrix p. Conversely, to every matrix 17 of size I x s (that is, with 1 rows and 5 columns) with elements from A, there corresponds an A-mapping, p: A‘—+A', which is obtained by multiplication on the left by this matrix. We shall refer to matrices of this type as A-matrices. To every A-matrix p of size txs and to every module it there corresponds a mapping p: ¢‘—>¢‘, which consists in the multiplication of the columns W on the left by the matrix. The image of this mapping will be denoted by pd)l and the kernel by dip. An A-module M will be said to be finite if it has a finite basis. For every integer k, for example, the module A" is finite, since it has a finite basis {2,}. This module is, moreover, free, since its canonical basis {42,} is formed of elements linearly independent over A. All finite modules will be endowed with the discrete topology. The free resolution ofa finite module M is an arbitrary exact sequence of the form ..._,ASz__£I_,A-II_PL,Asn_",M_,0‘

where p,, i=0,1,2,

(1.3)

are A-matrices. If the ring A is a Noether ring,

then every finite A-module has at least one free resolution. From now on, we shall assume always that the ring A is a Noether ring. This implies that an arbitrary submodule of a finite A-module is also finite. Clearly, a factor-module of a finite module is also finite. 1°. The tensor product of modules and the functor Tor. Let M be a finite A-module and let (I) be a free resolution of it. Let (D be an Amodule. In the sequence (1), we replace the powers of the ring A by the same powers of the module «p and we consider the sequence

~~v—+¢“L>¢“At¢’°—>0. This sequence, generally speaking, is only semi-exact, since the exactness of (1) implies p, pl +1=0 for all i. We consider the homologies of this

§3. Operations on modules

41

sequence, which are denoted by

M 8) ¢=Toro (M, o)= awn/po o" Tor.(M.¢)=¢, I-l mow

23

i=1,2,....

("l

In order to validate this notation, it is necessary to show that the factormodules do, in fact, depend only on the modules M and d) and do not depend on the choice of the free resolution (1).

Proposition 1 2. Let v~-—>A"1>A" AAHDAMIHO

(33)

be a free resolution of the finite A-module M’, and let I: M —>M’ be an A-mapping. There exist A-matrices f.., i =0, l, 2, ..., which make the diagram:

fiA"AA"fl>A”°—‘>M'~>0

A]

[.1

hi

[I

(4.3)

.--—>A"£»As Ante—H M—>0 commutative. If g, i=0, 1,2, ..., are other A-matrices which make this diagram commutative, then these two diagrams are homotopic, that is, there exist A-matrices hi: A"—> A"*',i=0,1, 2, , such that

fi—g,=n,h,+hi_l p,_1,

i=0, 1, 2,

(h_1=0).

(5.3)

Since the diagram (4) is commutative, the mapping fi: GW—MP‘" carries the submodule dim” into spud, and the submodule “45"" into 1|:i 45"". We can define the associated mapping

1“,: Q’v/pidi‘m—bdi li—l mow,

i=0, 1,;

(6.3)

Proposition 2. I. The mappings (6) depend only on the module 45 and the mapping f. II. Assume that the mapping f is an isomorphism. Then, the mappings (6) are also isomorphisms, and their inverses are mappings associated with some A-matrices. III. We again assume that f is an isomorphism. Then, if some mapping ni: ¢““—>(P" is a homomorphism, the mapping pi: 45"“—H1>“ is also a homomorphism. Proofl If gi is another matrix which renders the diagram (4) commutative, then by Proposition 1, the relation (5) is valid. But it is then clear that the mappings of the form (6) associated with the matrices f, 2 The proof of this proposition can be found in Canan and Eilenberg [1], Chapter V, and in Godement [1]. Chapter I, § 5.

42

l. Homologicnl Tools

and g‘ coincide, since the restriction of the mapping h,_I p,_l on 45,,” is null, and the component nih, carries the module 115, into 1ti W'“. This proves the first assertion. We assume that f is an isomorphism. Then Propositionl allows us to construct A-matrioes ¢‘ such that the following diagram is commutative:

—»A’2—Pl>A" ILA'o—tMfio ”I a] «I I!“

(7.3)

QAVILAvnfiAaofiMlao

and, therefore, the mappings $i: Qua/WWW“¢pi.,/Pi¢"”‘t

associated with the matrices 45,. are defined. We shall show that these mappings are the inverses of the mappings (6). Combining diagrams (4) and (7), we obtain the commutative diagram:

-~~—>A‘°A>A"AA‘°—tM—>0

My “.1 wot

u

-~-—+A“#>AIIAA’°—>M—>0 On the other hand, we can oonstmct an analogous commutative diagram, in which instead of the mappings ¢,.f,.. we find the identity mappings e: A"—t A". By Propositionl. both of these diagrams are homotopic, that is, ¢ifi—e"=p,h,+hi_lpi_l,

i=0,l,Z...,

(8.3)

with some A-matrim hi: A"—vA"“ (h_1=0). It now follows that the mapping éifl: ¢p‘V./pi¢n‘l—’¢p;.I/pi¢’hls

associated with the matrix ¢i fi coincides with the mapping associated with the matrix e; i.e., it is an identity mapping. From the symmetry

of the mappings d),- and f..,it follows that the composition 1’; if, is also an

identity mapping. This proves the second assertion. We now take up the third. Suppose that for some igo the mapping m: é" *‘ —> 4’" is a homomorphism, that is, that there is defined a manyvalued mapping P1 ni¢lnt_,¢asu’

inverse to 1n. We consider the mapping r: pidsnuintynu

I' My“. Gun fin.

§3. Operations on modules

43

Since the diagram (7) is commutative,

p,(r—h‘)=pi ¢t+1 Pfi‘l’i hi=¢lnipfl—plhl=¢lji_pihl' According to Eqr(8), the right-hand side is equal to e+h,_,p,_,. Since the restriction of the mapping hm p,_l on “45"” is null, the mapping “(r—hi), on pidir‘m, coincides with e; i.e. it is an identity mapping. Therefore the mapping r~hi is inverse to p,. Accordingly, p,- is a homomorphism. I] We shall assume that M'=M, and f is an identity mapping. Then according to Proposition 2, there exist A-matrices,f,, i=0, 1, 2, ..., such

that the associated mappings (6) are isomorphisms. Then the factormodules (2) do not depend on the choice of the free resolution (1), and this validates our notation. Let us determine the modules (2) in the case M = A" for some integer k. For the free resolution of the module A", we may choose the sequence 0—»A"—> A"—> 0. From the definition, we are led quickly to the isomorphisms

A‘ 8) «>249,

Tori(A", ¢)=o,

i; 1,

for an arbitrary A-module iii. In particular, for arbitrary integers k and I, A" (8 Al '5' A“.

Iff: M —>M' is a mapping of finite A-modules, then by Proposition 2 we can define the mapping Tori(M, ¢)-—»Tor,(M’, 45), depending only on f and a). Let (b: d) —> (P’ be a mapping of A-modules. It is clear that it carries dip” into 45;,H and pit?“ into piIP’m'. Therefore, there is an associated mapping

'Ppi_,/p,-‘P‘"‘—’4’L._./P.-4”“, and a corresponding mapping Tor..(M, 45) —> Tori (M, 45'), both independent of the choice of the resolution (1). If we have simultaneously the mappings f and 45, we may consider the composition of the mappings M ® 11> —> M’ ® 45 —> M‘ ® 45" For this composition, one employs the

special notation: [A', then their tensor product p®qz A”—> A" corresponds to the A-matrix p®q, formed as the Kronecker product of the matrices p and q. Proposition 3. Let M and L be arbitrary finite A-madules, let (1) be a free resolution of the module M, and let

->~—>A"—">A‘—"—>L—v0

l. Homologieol Tools

44

be a free resolution of the module L. Then we have the isomorphisms

M®L;A"/[poA"+qA"],

t=so, s=sl,

T0170", L); [Pa—1 A"""qA""']/(Pa_1®4) A“.

.

1:1,

(9.3)

where for brevity we have written 12,: pi (8) e‘, q =e‘ 8) q. Proof. We fix an arbitrary i g 1 and we consider the following diagram of type (y):

0

I 4

0

gm

m

Ann!

P!

1 ”l l

Ln

n-r

EH

A“!

“-1

pi—lA,”

qAnv

Pi-t

(pi_l®q)Alur

"l l

’0

i0

It is obviously commutative and the first and second columns and the second and third rows are exact. By Theorem 1 §2, we conclude that the module

LpH/p; B‘“ Eroriw, L) is isomorphic to the kernel of the mapping pi‘lA"‘—"> L“-I factored with respect to the submodule (pi,l®q)A"". This kernel is the intersection of the submodules pi_l A"'cA"'" ' with the kernel of the mapping b: A‘H‘—r D". The latter is clearly equal to 1124"". Thus, the isomorphisms (9) with i=1, 2, have been validated. Finally, let us consider another commutative diagram of type (y):

o (I) L' —”> L' —> 0 n b [in—P, A" fiAIK/pAsrfio

qA"'—> [qA'°+pA"]/pA"—>0

§3. Operations on modules

45

It is easy to see that the second and third rows and the first and second columns are exact. Therefore, we may apply Theorem] §2, whence follows the first of the isomorphisms (9). [I Since the relationship (9) is symmetric in M and L for the module M (8 L, we have the natural isomorphism M (8 L; L (8 M; in other words, the tensor product is commutative. The modules Tori(M, L) have the same property for all igl. The proof of this fact is left to the reader (it is suggested that he make use of the diagram (21.9) of Chapter VII). For arbitrary finite modules M, M’ and the module ¢ there is defined a natural isomorphism Tor,- (M e M’, 45)§Tor,.(M, (P) 6 Tori (M’, di).

This property of the module Tor, is called distributivity.

2". Flat modules Definition 1. An A-module 45 is said to be flat, if for an arbitrary exact sequence of finite A-modules M’—’»M—‘>M"

(10.3)

M'®o—ffl»M®oA'>w®o

(11.3)

the sequence (where I is an identity mapping of (P) is exact. If (P is a flat module, then, as is easily seen, Tor,(M, ¢)=0 for all i; 1 and for an arbitrary finite module M. It follows from Proposition 5 that this property is sufficient for the flatness of the module. The simplest example of a flat module is the module A“, where k is an arbitrary integer. In this case, the sequence (11) takes the form:

[MT—M” [Mr—M” [M"]‘. Therefore, the flatness of (10) immediately implies the flatness of (11). Other examples will be met later. We note some properties of flat modules. Let ¢ be a flat module and let p: A“ —> A' be an A-matrix. Then, if the sequence A‘ —"> pA‘ —» 0 is flat, the sequence

A’®d§—”§l—>pA‘®¢a0. is algebraically exact. Because of the isomorphism A‘®¢;¢‘ and the monomorphism p A' (8) 45 a A' ® 45 z W, the mapping p (3 I is isomorphic to p. Taking account of the fact that this mapping is an epimorphism, we arrive at the isomorphism pA’®¢;pd5’. Let N be a submodule of the finite module M. The identity imbedding N —> M is a monomorphism, and therefore, for an arbitrary flat module

I. Homological Tools

46

(P the mapping N®¢—>M®¢ is also a monomorphism. Therefore the modules N ®d> can be looked on as a submodule of the module M (3 45. We note that we have the isomorphism M®¢/N®¢;M/N®¢.

To prove this, it is sulficient to consider the exact sequence 0 —> N —> M —> M/N —> 0 and, subjecting it to tensor multiplication by the module @, to make use of the fact that it is flat. Proposition 4. Let M be an arbitrary finite module, let K and L be submodules of it, and let 45 be a flat module. From what we have said above, it follows that the module (KnL)®d) can be considered as a submodule of the modules K®¢ and L®¢. We have also the equation

(KnL)®¢=(K®o)n(L® 4:). Proof We consider the sequence 0—»KnL—‘rKQBL—’>K+L—»0, where a is a mapping, carrying the element x into the couple (x,x), and the mapping p carries the couple (x, y) into the difference x — y. It is clear that this sequence is exact. Therefore, if we take the tensor product with the module ‘15, we arrive again at the exact sequence

0—»(KnL)®¢L°'.(K$L)®¢P+°'.(K+L)®o—.o. (12.3) Because the tensor product is distributive, we have the isomorphism

(K e L) ® 452(K ® 4))63 (L® 4)). We substitute this isomorphism in (12). It is clear that the mapping a ® I goes over into the mapping a, and the mapping p (2) I goes into the mapping p, that is, we obtain the following exact sequence: 0—v(KnL)®d5—'—>(K®d§)®(L®¢)—’r(K+L)®(Ii—>0.

Since it is exact,

0(ao=1ma=Kerp=(K®¢)n(L®q>). n 3°. Criteria for names of modules PropositionS. In order that the module (15 be flat it is necessary and suflicient that, for an arbitrary exact sequence oftheform A’ J» A' —'> A", the sequence 4" —'v 415‘ _', 115” be exact.

§3. Operations on modules

47

Proofl The necessity is obvious. We shall prove the sulficiency. To begin with, we establish a formally simpler proposition: the exactness of a sequence of the form

0., M'—f——> M—‘» M”—»0

(13.3)

implies the exactness of the sequence

04M'®¢£>M®¢P'+°I>M”®d§—>0.

(14.3)

Lemma3. The sequence of (13) can be included in a commutative diagram of the form:

.J.——»A=I—>"° A‘°——>‘ M—>0

(15.3)

“glissilglso l

l

l

0.

0

0

Subjecting diagram (15) to tensor multiplication by d}, we obtain the commutative diagram:

0

o

o

l ,,l ”l“

o

l

..._.qp"z'—Z'—.¢Fi'A>¢'"—>M"®¢——>0

4 4 4 4 I Al «I, 4

..._,¢)r'zlnp'I—E°—»d>'°—»M®¢ —+0

(16.3)

——»d>"—’°+¢'°—>M’®¢P -—.o

l 0

l 0

l 0

[[1, 3 The proof of this proposition can be obtained from Cartan and Eilenberg Chapter V w. by replacing all projective modules by free modules.

48

I. Homological Tools

It follows from the hypothesis that all the rows, and all the columns with the exception of the right-hand column, are exact. Applying Theoreml of §2, we find that the right-hand column is also exact; ire. the exactness of (14) is now proved. We now establish the exactness of (11) under the assumption that (10) is exact. The fact that (10) is exact implies that the sequences

0—»Coimf—I.M_l.1mg—.o, 0—»Kerf—»M’—>Coimf—>0—>Img—rM"—»Cokerg—>0.

are exact. But then the proposition that we have just proved implies that the following sequences are also exact:

0—»Coimf®¢—>M®¢—>Irug®¢-»0, 0—»Kerf®¢aM’®¢—>Coimf®¢—s0, 0—»Img®M"®d§—>Cokerg®¢—>0, whence it is not difficult to derive the exactness of (11).

U

4". The functors Horn and Ext. Let 4} and M be A-modules. We consider the set of all A-mappings from M to 45. Such mappings can be combined by addition and multiplication with the elements of the ring A in the customary way. It is clear that the set of all such mappings with these operations is an A-module. This module is designated by Hom(M, «15). If we are given a mapping f: M —>M’, then there is defined the mapping Hom(M’,¢)—> Hom(M,d5), which relates to every element FeHom(M', 45) the mapping M axaF(f(x))e¢. This mapping will be denoted by Hom(12¢) or by f‘. Let k be an integer. We consider the module 45". To each of its elements (c151, ~~~,¢x) we associate the mapping A‘a(al, ...,u,‘)—»Zai¢.-e¢.

This correspondence defines a mapping h: 45‘—>Hom(A",¢). Proposifionfl The mapping h is an isomorphism For an arbitrary A-matrix p, the following diagram is commutative:

Hom(A',¢) "“M‘M’ Hom(A',d>)

I. v

where p’ is a transposition matrix.

I» 1”

(pl

(17.3)

$3. Operations on modules

49

Proof. First we note that the mapping h is a one-to-one mapping. In fact, if 11(4),, . t. , 03k)=0, then all the ¢i=0, since the module A" consists of columns of the unit matrix. We construct the mapping h“. Every mapping F: A" —-> d) is uniquely defined by the values F(e,), «=1, ...,k, on the basis elements; it can be written in the form F(a)=(F, a), where F is a vector formed by the elements F(e,). Considering this vector as an element of the module 45", we obtain the mapping h“: Hom(A‘,d5)—>d5". The composition hh“ is clearly an identity mapping. Therefore, h is an isomorphism. Let us go over to diagram (17). By definition, the mapping Hom(p, Q) carries the element FE Hom (A', 45) into a mapping which acts according to the formula F’: a—>F(p a). Making use of the isomorphism h, we represent this mapping in the form

(h“(F'),a)=F’(a)=F(p a)=(h“(F).P a)=(P’h“(F),a). It follows that h“(F’)=p'h"(F).

[I

The functor Horn can be defined in another way — by analogy with the functor of a tensor product. In the free resolution (1) of the module M, we replace the modules A‘ by the modules 45“, and the matrices Pk by the transposed matrices p3:

OAWLW‘J‘AQ‘Z—wu. This sequence is semi-exact, since the exactness of (1) implies p; p£_1=0. The homologies of this sequence will be denoted as follows:

Ext°(M,d>)=¢mv,,

Ext‘(M,tP)=tP,;/p;-_1¢""

i=1,2,

(18.3)

We shall show that there exists an algebraic isomorphism Ext" (M, 4)) _—"_' Horn (M, db).

(19.3)

To every mapping F: M —> 45, we set up the mapping F’: A‘“a¢, which is the composition of F and of the mapping a: A‘°—rM from (1). The composition of the mapping F' and the mapping p0: A" AA“ is clearly null, that is Hom(po,¢)(F’)=0. If we apply to F’ the mapping h“. we obtain an element f64”“, and, according to Proposition 6, p}, f =0. Conversely, to every element f545,3, the isomorphism It sets up a correspondence with the mapping F’: AW—HP, which reduces to zero on the submodule 110 A". Such a mapping can be looked on as acting from the factor-module A‘D/po A‘IzM to d). The isomorphism (19) is then established. Let us validate Definition (18). Let f: M —> M’ be a mapping of finite A-modules, and let (1) and (3) be free resolutions of these modules.

50

l. Homologieal Tools

The commutativity of diagram (4) implies that the transposed matrix fi’ carries 45,1 into 45”, and carries 1I:;-_l WM into p3_1d>"-'; therefore, there is defined an associated mapping

1?: unmade-I» ,5/p2-1¢~-a

(20.3)

In the same way as we proved Proposition 2, we eon prove the following

Proposition 7

I. The mappings (20) depend only on the mappingfand the module (P. 11. We assume that the mapping f is an isomorphism. Then the mappings (20) are also isomorphisms, and their inverse: are mappings associated with certain A-matrices.

III. Letfbe an isomorphism. Then iffor some i the mapping Hom(n,—, '15) is a homomorphism, the mapping Hom (pm?) is also a homomorphism. It follows from these propositions that the modules (18), up to an A-isomorphism, depend only on M and (D, and iff: M —> M’ is a mapping of finite modules, then the mappings

ff: Ext‘(M’,¢)—>Ext‘(M,4P)

i=o,1,2,...,

are uniquely defined, and f0" =f*. Let 4:: iii—HF be a mapping of modules. It carries the submodule ‘Pp; into «Egg, and carries the submodule p§_,’"‘ into p;_1(15""'. But this defines the associated mapping dz}: Ext‘(M. it) a Ext‘(M, 45'), which is easily seen to depend only on 4) and M. The mappings 45;? and f3 will be called derived (with respect to 4: and f).

5". Exact sequences connecting derived mappings. We have in mind the well—known exact sequence for the iunctor Ext. Let «P be a module and

OHM—hM—hM'go

an exact sequence of finite modules. Then there are defined the socalled connection mappings 6;, 1:0, which make the sequence

04 Hom(M”, o) i» Hom (M, o) J; Hom(M',«p) 3» Ext‘(M”, o) —»

_. Ext"(M", III and ¢—> 1. A detailed study of these operators, which we carry out here, will be extremely important in Chapter IV. § 1. The space of power series 1°. Terminology and notation. Let s be an integer. The elements of the space C‘ will often be interpreted as columns of height s with oom— plex elements. For an arbitrary Ce C‘, we write

|£|=m3XIC,I,

{=(C.....,Cs).

Correspondingly, the linear operator a: C‘—» C’ will be identified with the matrix a= (aw) of size txs (that is, with t rows and s columns) which acts on the column vector 2; by multiplication on the left. We write

|a|= 152;: max ,_1 2 lawl. We have, obviously, In“ §|a| ‘ ICI for arbitrary CeC‘. Proposition 1. Let a be a non-singular matrix of size sxs. Then s I a Is— 1

Ia"|_S_ |det al '

(1.1)

Proof Let A”, be the cofactor of the element a”. It is clear, that Mulé H Zlar-a-Iélal'_‘~ ¢'*r 11'

Therefore the absolute value of each element of the matrix a" does not exceed lal“l |det a[“‘. From this follows the inequality (1). [I Sometimes it will be convenient to use the following definitions. Let M and N be arbitrary finite sets; a matrix of size M xN is defined as an arbitrary function A = (a, v}, defined for ueM, veN, with complex values in the field of complex numbers. The set of numbers a“, for fixed ,1 will be

54

ill Division with Remainder in the Space or Power Series

called a row of this matrix, with the index ,u, and the set of numbers a“, ueM will be called a column with index v. By enumerating the elements of the sets M and N, we will convert the matrix A into the customary

rectangular matrix. From this as a point of departure, we proceed to define the concepts of minor, rank, and so on, for arbitrary matrices of

dimension M xN. Let A=(a,"} be a matrix ofdimension MxN; if M’cM and N’cN are subsets, the set of numbers {auw yeM’, veN’) will be called a submatrix of the matrix A. If A is a matrix of size Mx N, and B is a matrix of size Nx L, we can define the product AB, which is a matrix of size M xL.

Let us now choose an arbitrary integer m. By Z"I we denote the subsets of the Euclidian space R'", consisting of points with integer coordinates, and by 2:, the subset of 2", consisting of points with nonnegative coordinates. The sets Z"I and Z"; will be called Z-modules. The vectors e,“ k=l,...,m, forming the rows of the unit matrix of order m, will serve as a basis for the Z-module Z'”. In Z"1 we now consider the linear functional ialil, which maps each point on the sum of its coordinates. The intersection of Z"; and the set in which |i| =kg0, will be denoted by 2*. We define an ordering relation in the lattice Z”I as follows: We write 1‘ gj, if no coordinate of the point j excwds the corresponding coordinate of the point i, that is, if i—jeZfi. We shall write i>j. if igj, and i$j. Let s be an arbitrary integer. We shall denote by 5 Z’" the set consisting of all pairs (0, i) where a is an integer with values in the range from 1 to s, and i 62'". More generally, if I is an arbitrary subset of Z", we shall mean by 5.! the subset of :2” formed of pairs (0, I"), where a: 1, ...,s, and is]. The set 11 is identical with 1. Given any subset

JcsZ'" we denote by III the number of its elements. If I is a subset of 52"“, and I is a subset in Z“, we denote by J +1 the subset of 52’", consisting of the pairs (0-, i+j), where (a', 06!, and j e].

2". The space of formal power series. We now fix the Euclidian space 0"; the points of this space will be denoted by "=0“, , r1”). By .9’=£/’[n] we denote the linear space consisting of the formal power series in the variable 11, with coefficients in C, that is, the space of formal

sums of the type

¢=¢Dfl=§w¢mi

i=0},

13.). 'I‘=nl' mnl'r-

(2-1)

For an arbitrary integer s we denote by 37” the direct sum of s replicas of the space .9’, in accordance with the notation of §l of Chapter I. We shall interpret the space 5" in two different ways: first, as the space of columns of type 5, made up of elements of 5’; and, on the other hand. as

§ 1. The space of power series

55

the space of formal power series in r], with coefficients from C‘. In the latter case, we shall write the elements of the space 5/" in the form of the sum (2}, to remind us that ¢,.e C' for all i, or in the form ¢= 2 ¢a.i'fa'i3

¢a,i€c.

sZ'S'

where r,“ is the product of r" and the column e, with the index a of the unit matrix of order s. For every pair (a, i)ex Z': we consider the linear functional 6“ over 5’". defined by the formula 5m: ¢=Z ¢:.j'l"]*’ ¢a.i‘

We set 6,, i=0 for iEZZ. If i621, then 6; will denote the functional over .9" with values in C‘, which maps the series (2) into its coefficient 43;. Thus, the functional 6; may be considered as a column of height 5, consisting of the functionals 6“, a: 1, ..., s. More generally, let I be an arbitrary finite subset of 52'1- By 6, we denote a functional on Y‘, defined by the formula

6,: 4:» w...“ (a, owed“. In other words, 6,¢ is a column vector, formed from the coefficients of the series 4: with indioes belonging to the set .9“. We shall write simply 6,, for the functional 6,“. Let s and t be arbitrary integers. By a functional over .9" with values in C‘, we shall mean an arbitrary finite sum of the form

(3.1)

f=f(6)=Zf"“5.,nf"'eC‘, which is defined by the formula

f5 ¢->Zf"'i¢a,.-~ In the matrix notation Eq.(3) appears as follows: [=2 f‘ 6‘, where f‘2 C‘—> C' are matrices consisting of columns f" '. The maximum value of lit for f‘ +0 will be called the order of the functional f, and will be denoted by deg f The differential operator, corresponding to the functional f, is the 1 expression

{(D)=z f"‘—5 ow; ¢alil

where D”'i is the product of the operator 0‘: anthem”: and the column e, with the index a of the unit matrix of order s. The following

56

ll. Division with Remainder in the Space of Power Series

formula connects the functional f(5) and the operator f(D):

f(5)¢=f(D)¢l.,-o~ 3". Operators in the space of formal series. We again let 5 and t be arbitrary integers. By an operator acting from .9” to .9", we shall mean an arbitrary expression of the form

A= Z 71"‘At.i(5), as

(4-1)

where all the A,_,- are functionals on .9" with values in C, as defined by ‘ the formula

A1¢—‘Z'I"Aa,i(5)¢For every i, the column A,=(Am, ..., A“) is a functional on y' with values in (3'. Therefore, formula (4) can be rewritten as: A=Z "[Ai(6)‘ But this formula can again be rewritten, if we express the functional A‘ in expanded form

A=Zn‘A!5;=Z'I"‘AZ.'{6..,-, where Al: 0—» C’ are matrices, consisting of the quantities [43;]. The quantities A1, A{, and At! will be called the coefficients of the operator A. They may be determined from the operator itself as follows: Ar=6iA,

A{=6,An’,

A:'/=5:.1A'la'l~

In particular, the identity operator in .9" can be written in the form: E=Zfliéi=ZWM5m~

We shall mean by [-],, an operator in 9" which acts according to the formula . [¢]k= 2 ’1I 6-415High

The kernel of this operator will be denoted by "1;“. Thus, mg”, is the space of power series in 5”, not containing terms of order gk.

4°. The space of convergent power series. For every r>0. we shall consider the subspace 0,0, we have the inequality

m I” . W44) mu zen.

(5.1)

Proofi We denote by 4501) the sum of the series 2 d), 71‘. This function is bounded on the sphere |n|§e and is analytic within the sphere. We

inscribe in the sphere the polycylinder {r}: |m|§%, i=1, ...,m} and m we apply Cauchy’s theorem, taking as the contour of integration the skeleton F of this polycylinder. Then a chain of self-evident relationships .r Iil 1 ‘ l¢z|= —D'¢ =|(27r)‘"',_l'rl“"¢(71)d'l §(@) l|¢ller 9:;el: 1‘! "=0

leads us to our goal.

I]

5°. The spaces 9’ and (D as Q—rnodules. We fix an arbitrary integer ngm and consider the ring 9: C [2] of plynomials with complex coefficients in the space C": C2. The space C“ introduwd earlier will now be looked on as a coordinate subspace in C", or more exactly, the point 716C” will be identified with the point z=(r,l, m, 7],," 0, ,0). Corresponding to every polynomial f5.? and point 250”, we determine an operator in .9’

f(2)= ¢['I] *f(1+'1)¢['1],

(61)

whose action consists in a multiplication by the power series Di

f(2+r1)=Z 'l'i—INZ),

which is the Taylor expansion of the polynomialfat the point z in terms of the variable 71. Similarly, corresponding to every 9-matrix p: 9’»? (that is, a matrix of size t>P(Z+'I)¢[VI]6‘9", which acts by multiplication by the Taylor expansion of the matrix p. Let us fix the point 2. By setting up the correspondence between the polynomial feP and the operators (6), we have converted the space 5’ into a unitary 9-module. The space V, with this structure of a ?-module, will sometimes be denoted by 9;. The submodule of this module consisting of elements of the subspace 0 will be denoted by 0,. When n=m the submodule will often be identified with the space of functions holo— morphic in the neighborhood of z, where the polynomials of the ring 9 act in the usual way.

II. Division with Remainder in the Space of Power Series

58

In what follows, if we do not explicitly state the contrary, the spaces .9’ and (0 will be given the structures of the modules 5’1, and 00. We note a few well-known properties of these modules‘. Proposition 3. When n=m, the 9-modules 5” and j or i =j. We remark that the relation i>j is preserved under transfer in the lattice Z’". In accordance with the definitions of § l, 132'" is the set of pairs (1', i) where ieZ’", and r is an integer not exceeding t. In this set, we also introduce an ordering relation, writing

(T.i)>(¢f,D,

if i>j or i=j and T(¢rafl or (r, i)=(a, j). In the case (1', 0207,j) ((1, i)>(a,1)) we shall say that the point (I, i) is (strongly) senior to the point (a.j). Let A=(o:1, , 01,) be an arbitrary finite subset of tZ'"; we shall say that it is ordered by seniority if al>m>zxr Let A=(rx1, ...,u,) and B=(fi,, ...,B,) be two subsets of tZ"l with the same number of elements, and let them be ordered by seniority. We shall say that the set A is strongly senior to B if for some kgr, we have the relations “i=fl1n ”’tak~l=flk—l’ mk>fik~ We shall say that A is senior to B, if A is strongly senior to B or if A =8. Let us now fix an arbitrary point 260' and consider the base series A," k=0, 1,2, ..., at this point (see 1°). For every k the rows

3:.ifl=(5z.tfi.i-~~~a5:.ifl,;k).

(wet/Ct,

form the matrix 41,. We fix an arbitrary k and we consider all possible sets 1:22}, having the property that the corresponding rows 6“; 1;, (1.06}. of the matrix A,‘ form a basis in the linear space spanned over all rows Ak. The number of elements in each such set ,1 is equal to the rank p. of A," Ordering each of the sets 1 by seniority, we indicate the most senior by It. This set is characterized by the fact that IJII 2 pk, and an arbitrary row EUR of A, is a linear combination of rows of the same matrix with indices 0 and |i|_;—K imply that for some ku.,

But this shows that the row 6,, I} of the matrix A x with the index me],K is a linear combination of rows of the same matrix with strongly senior indices. This is impossible because of the property of the set JK. We have obtained a contradiction which proves the inclusion relation I. We now take up the proof of the second assertion. Let M be a nonsingular minor of the matrix A} of order p. For simplicity we shall assume that it lies in the first p columns of the matrix. We consider the matrix .1! of size sxxp, of which the first p rows consist of the matrix M ", and the remainder are equal to zero. We write D=AK.ll;

D"=A{(.l(,

jeZK.

Let us determine the structure of these matrices. D is a matrix of size tixxp, and its minor, formed of rows with indices belonging to I“, is the unit matrix. For every aetEK we denote by d“ the row with index or. For every :1 the row dI is a linear combination of the rows (in, with indices a,eJK which are senior to (1, since the matrix AK has a similar property. Let M}, ..., at, be elements of the set J“, all senior to on Then only the first 0' elements of the row 11, can be different from zero, since d“, , d,” are the first 0 rows of the unit matrix. Further, let d”, «:1,

p be the

column of the matrix D with index a. From what we have said already, it follows that all the elements of the column 11" with an index strongly senior to at, are equal to zero, and the element with index at, is equal to unit . ¥he element of the matrix D(Dj), located in the column with index aand the row with index at, will be denoted by dfldfl'"), Relation (13) implies that . d“ ._ if igj, M: “‘ 1’ 14.2 d' {0 otherwise. ( ’ Thus the connection between the matrices D and D’ is the same as that between the matrices A x and Ag, and in fact, the mmor of the

66

II, Division with Remainder in the Space of Power Series

matrix 0", made up of rows with indices lying in :2} +j is equal to D; the remaining rows of the matrix D’ are equal to zero. Let us consider the system of equations

(15.2)

2 Dix}=y, 162.:

where x36 2”, the solution of which yields also a solution of the system (9). In expanded form, this system reads as follows: d""x-,=y, I-

where

(x-In ,,...,x;I V)=x},

(16.2)

and d“ are columns of the matrix Di. We now show that there exist elements ximeg’, satisfying these equations with indices belonging to the set Jx+£m and that these elements can be written in the form

(17-2)

Xi..=2f}_.y.~,

where rj‘JEC‘ are rows, made up of polynomials in the elements of the matrices AK and M ‘1, and are different from zero only for Mi); Mi, +j) and satisfy the inequalities

Ir},al§bl>"“"“"+”"'.

bo=lix| t lD|'-

(18.2)

The x“ will be determined by induction on the value of the sum 1', +j. We assume that for some i625 we have determined all the quantities xj', with i, +j>i, in the forms (l7)—(18) above, in such a way that all the equations in the system (16) having indices (1’, i')EJK+£,‘ such that i’>i are satisfied. We remark that the values of the xiv, not yet determined cannot disturb the fulfillment of these equations. In fact, it follows from the formula (14) and the properties of the 11:"; that the most senior, non-zero element of the column d“ has the index (1,, I", +j). Therefore, if i, +jgi, all the elements of the column 11"” with indices (1:,i’), where i’>i, are equal to zero.

We now need to find quantities xiv, with i,+j=i to satisfy all the equations of the system (16) with indices (1:, fleJK+Zx (where i is the point fixed earlier). But this means that the equation 2 diiixjm=l7mu i,+]-i

Where

9r.i=Yz,r—

Z d{:'i"xj’,u' 56,”).-

(19-2)

has to be satisfied for all (I, {)9}K + 2x . Taking aooount of the relation (14) we rewrite the system in the form:

M2] 43.1. x]..=fi.,i-

(20.2)

By definition of the set JK+ZK, there exists for each of its points (1,1) at least one pair (6,1), for which [gag p, and 1'e is such that i: i, +j,

52. The base sequence of matrices

67

and r = 1,. If to the given point (t, 0 there correspond several such pairs (0, j), we select exactly one such pair (for example, by the condition that the point (a,j)ep£,( be the most senior) and for the remaining pairs (0‘, j), we set xj‘,=0. Thus, the number of undetermined quantities xj‘, in the system (20) is equal to the number of equations in the system. Rearranging the terms on the left side of (20) in order of increasing 1,, we transform the system into upper triangilar form. In fact, for arbitrary 1(t,, i,)=ac,. On the diagonal of this triangular matrix we find the quantities 11;, i,=d;',, which are equal to unity. Thus, the determinant of the matrix of the system (20) has value unity, and accordingly, the norm of the inverse matrix, because

of the inequality (1.1), does not exceed t IDI“ ‘. Therefore, the system (20) has a solution, which can be written in the form xj,a=P§,a§n

ii=@l,h"-)fir.i)s

(21-2)

where p3,, are rows made up of polynomials in the elements of the matrix D, satisfying the inequalities

(22.2)

Ip}..l ét IDI““

We shall verify that the x17, that we found has the form (l7)—(18). For this, we substitute in (21) the expression (19) for the f), in vector notation , ’

x,,,=loj~...ys-p§..,i 225:1!” x”

” =p§..yi-p;,. z diyZdimi,:+j’>l'

I‘

‘23.2’

We shall show that in the second term of the righthand side, the summation extends only over the points i' satisfying the inequality L(i’)> L(i). In fact, the diff. are different from zero only if i—j’EZKr We conclude that |i,,.+j’—i|_ glj’—i|_ ; —K. Since i,l+j'>i and |i,,.+j'—i|=0, then Propositionl implies that L6,, +1")>L(i). 0n the other hand, it follows from the induction hypothesis that rfi,.¢0 only when L(i’);l.{i,,y+j’). This inequality together with the preceding implies that L(i’)>D(i), whence, in particular, it follows that i’*i. Therefore, we can convert Eq. (23) to the form (17) by writing

r}..=»5-..,

2 47-1"? ri’.= —p‘,.. 3944'):

(24.2)

where i =i, +j, and L(i’)>L(i). We remark that the elements of these '1, matrices are polynomials in the elements of the matrices A K and M since the matrices pi, have this property, and by the induction hypothesis, so also do the matrices ’52,, with if +1“ >i.

11, Division with Remainder in the Space of Power Series

68

We estimate the norms of the matrices r}:,,. When 1" = 1', the inequality

(18) follows from (22), since tlDl"‘§bo, and L(i)=L(i,+j). For i’#i we make use of the second of the formulae (24). The number of terms on the right side does not exceed the number of points j, for which dfi'_ ,w,*0 and, therefore, does not exceed the number of points 1", for which 1—} EEK. But this, clearly, 15 not greater than (2‘). Making use of this fact and the inequalities (22) and (18), we obtain Ir}:.| éll . t . |D|"‘ . |p| . bLLli’kLtiar-t-i'll” gbbumrmum‘“

since

L(i, +j')> L(i)= L(i, +j).

Thus, the formulae (l7)—(18) have been proved for all pairs (0, j). We remark that for arbitrary a. iozK e," li,|=K, and, accordingly, by virtue of Proposition I, L(ia); L(K em), whence L(i, +j); LU + K em). Therefore the inequality (17) can be rewritten in vector form x} = Z I; y;,

(25.2)

where r} are matrices different from zero only when L(i);L(j+Ke,,,) and satisfying the inequalities l’ilébii'm_l'u+xz'"’lfl~

(26.2)

Since the vectors x} satisfy the system (15), by writing xj=.I(x}, jeZK, we obtain a solution of the system (9). It follows from (25) that the vectors xi can be written in the form (10) with R}=Jlr}. From what we have said above, it follows that the elements of the matrices R;- are polynomials in the elements of the matrices AK and M ". and that the matrix R:- is different from zero only when L(i)gL(j+K em). In view of the inequality (1.1) the norm ofthe matrix .1! does not exceed )2 IAKI" l(detM)‘. Hence, taking account of the fact that 1n the inequality (26) the exponent is always positive, we obtain the inequality (11) for the matrix R‘ I] § 3. Stabilization of the base sequence 1". Monotone sets Definition 1. A subset J cZ': will be said to be monotone if with every one of its points i it contains all points jgi, or, in other words, if it coincides with the union U {i+Z'1, iEJ}. The set J=(J‘ tZ': will be said to be monotone if each of its components 1‘, . is a monotone subset of Z1. Given a set I CtZ'i and an integer kg0, we denote by I) the intersection } n t):,,. We can formulate the criterion of monotonicity for sets

§37 Stabilization of the base sequence

69

as follows: in order that the subset J :12"; be monotone, it is necessary that for arbitrary k;0 and arbitrary [:0 (and sufficient, that for all kgo and 1:1) we have the inclusion relation J,.+Z,CJ,,+,. Thus, in accordance with the Fundamental Lemma of §2, every basis set is monotone. Definition 2. The source of the monotone set J t'l is defined as the minimum subset 3(1). having the property that

J=sm+z~1. In other words, the source is the set of all points (I, Us}, having the property that there exist no points (t,j)eJ with j 0 and q. Proof. The varieties Nv will be constructed by induction We set N0 = C" and we postulate that for some go we have already constructed the variety N0, ...,Nn, satisfying the conditions of the theorem, with v=0, ..., y— 1. If the variety N, is empty, the theorem is proved. Supposing that Nu is not empty, we construct the variety NF,+1 so as to satisfy the conditions of the theorem with v=a We begin by constructing a sequence of sets Act 2,, k=0, 1, 2, ..., and a sequence of algebraic varieties L,“ such that

LoCLrC"‘CLrC"'SN,,

93. Stabilization of the base sequence

71

using the following inductive construction. We assume that for some kgo the sets 10, ...,/k_1 and the manifolds Lo, ..., Lb! have already been constructed, and for convenience we set ]_1=Q and L_ 1=13. Then 1) we determine max |J,‘(z)| for the point zeN‘\L‘_1, and among the sets J,‘(z) for which this maximum is obtained, we choose the most senior, which we denote by )1; 2) the union of Lk_1 and of the set of points zeNfl\L,_,, in which Jk(z)+ 1*, will be denoted by L,. It is clear that Lb] CL'ENI" It remains to show that L,‘ is an algebraic variety. By A} (2) we shall denote the submatrix of A,‘(z) consisting of the rows with indices belonging to I,“ We shall show that the equation J,‘(z)= 1,, for points zeNu\Lk,1 is equivalent to the equation

rank 111(2)= Iltl-

(3-3)

In fact, if .I,‘(z)= I. the rows of the matrix 211(2) (the number of these is equal to Ilkl) are linearly independent and therefore we have Eq. (3). Conversely, if Eq. (3) is satisfied, then by l) the rank of the matrix Ak(z) is equal to Ilkl. Therefore, by its construction, the set J,‘(z)3 is senior to ’6‘. On the other hand, by construction, the set 1,, is the most senior of the sets J,‘(z), for which |J,.(z)| =|},‘|, and zeN‘,\Lk_,. It follows that

Ju(2)=lt~

The condition contrary to (3) can be written in the form (4.3)

det M‘(z)=0, V,"

where the M‘(z) are all the minors of the matrix A742) of order lfil. By 1) and 2) the sets 10(2), ...,J,‘,,(z) are constant on N,\L,‘_1, and therefore, the ranks of the matrices 210(2), . .. , Ah! (2) are constant on Nu\Lk‘1. The theorem of § 2 implies that the matrix A.(z) has a constant size on this set and its elements are polynomials in 2. Therefore, all the det M"(z) are also polynomials on the set N,,\L,‘_,. Accordingly, the system of Eq. (4) determines an algebraic subvariety L’ cNu\L,,_,. It follows from 2) that L1, is equal to L’ UL~,I and, accordingly, is an algebraic variety, which is what we were to show. This completes the construction of the sets 1,, and LR. At an arbitrary point zeM\L., we have J,‘_,(z)=,,€,‘_l and J,(z) =1,“ From this, because of the monotonicity of the set 1(2), it follows that I,” 1 + 21 c I,“ which implies the monotonicity of the set I = U 1,, c tZ'l. [1

Let K =K (j) be the stability constant of I. We shall show that for arbitrary k > K the equation Lk= LK holds. Let us suppose the contrary: 3 See 3", 52.

72

II. Division with Remainder in the Space of Power Series

let I: be the smallest of the numbers for which this equation is not satisfied. In this case Lk is strongly senior to LK; we choose an arbitrary point zeLh\LK. It follows from 2) that J,(z)=}, for all i§K. Then by the properties of the constant K, it follows that 1(2) :2 1. 0n the other hand, it follows from 1) that |J,,(z)|§|/,,|, whence, employing the preceding inclusion, it follows that Jk(z)=}h that is, zéLk, which contradicts the choice of the point z. The contradiction that we have obtained shows that Lk=LK for kgK. Since cLK for k k, the assertion (3“) follows immediately from (3,,_1), since 1L1 git. We shall now suppose that mg); We first suppose that ko,

M—i+Ke,—Ke,,|§0,

Il—i+Kel—Kem|_;l—i|_> —K. Hence, in view of the proposition of § 2, there follows L(l— i+ K el— K em) 2 0, and this, because of the linearity of the operator L yields L(l.) ; L(i) — L(K e1 —Ke,,.)= L(n— It. If, however, |i| > k, we have

[t—ilgk—IiKO.

ll—il+éllléK.

which again, by virtue of the proposition of § 2_, leads to the inequality

L(t—ngan—k) e,_= 4;, whence magma—1;.

F more Now suppose k _2_ K. We make the construction of the vector precise. To this end, we rewrite the system (8) in the form

EALF,=6,‘Q"'1, 1

(15-4)

80

II. Division with Remainder in the Space of Power Series

where ALEX,“ K is the minor of the matrix A K formed from the columns 6,, vi Emu, 71), a: I, , 5K, and F} is a column formed from the unknown functionals F“, a: 1, .r., 5x which coincide up to numerical order With the functionals E, in (8). By the Fundamental Lemma of § 2, we can find columns Fj satisfying all the equations of this system with indices belonging to 1,, which admit the representation

(16.4)

13:2 R5595“,

where the R} are the matrices constructed in the Lemma, non-zero only for L(i’)g LU+ K em). Eq. (9) can now be written as:

git—92%": —6iZ'IjH F): —Zér-;flc2 $19.5“.

,-

,-

i‘

If we apply both sides to the element r", we obtain

9in‘-9i‘"rll= —Zr5z_,-R:ZR§595“VI“ i i'

(17-4)

We shall show that the values of the indices 1', 1", j, A, for which the terms on the fighthand side are diilerent from zero, are connected by the inequalities L(1)%L(IV);LU+K€..);L(i)-li‘

(13-4)

Since on the righthand side of (l 7) |i’| =k>k— 1, (3,,_ l) implies that 9?.“ rl‘$0 only if L().)gL(i’)+e,,. or if i’=}.i In both cases, the first inequality of (18) is fulfilled. The second follows from the fact that R} $ 0 only if LU); LU + K em). We now establish the third inequality. We note that 6i,jfl(+0 only for i gj. We assume that |i|=k. Then the equation |j| = k — K implies, that j—i+Ke,zo,

|j—i+Ke,|=0,

Ij-i+K€i|_;|j-ilc=-K-

Hence, by the proposition of § 7, L(j—i+Ke1)§0, that is. L(j+K em); Lti~Ke1+Ke,,,), which leads to the third inequality of (18). Now suppose |i|>ki Then |j+Ke,,,—i|=k—|il I, such that Vic ”am for arbitrary jeJ. We can then define the conjugate mapping 0*, which acts on cochains:

9*: ‘15:! 2 . ¢io.~-».I'v Um"

A Ui."¢lv=z$wntm.ou.) Vin"

A Viv)

where $000)..0w is the restriction of the function ¢gunl>nvgud on the region VlonmnVlv.

3°. Elementary coverings An elementary covering U of the space C”, is a covering consisting of spheres

U,=(C21C-Z|0, s>0, and q be real numbers. In the covering S, consisting of the spheres

S,={C: IC-Z|Ce ( Izl’+1 C39“(z,./V)=Ca(lz'1+

whence it follows that the sphere S“ with center at the point z and r’ radius C e (W )ll lies within the sphere S, Accordingly, for arbi-

trary ZEN“,1 the radius ofthe sphere Si" is not less than rv_l(lz| + l)“’"‘, where

rvil =minlrv 2’1"", CSDV 2—1""]"),

(33)

Pv—l =maX(p.. |q|(pv+2))-

Thus, the desired covering SV"=(S;“} has been constructed. If we carry out the induction, we constmct the covering S": {Sf}. If we de— crease the radius of each of the spheres S? to r0([z| + 1)"’°, we obtain a covering satisfying the hypotheses of the proposition. Since rm=Ce, and pm:2q, we find from the recurrence formula (3) that r0 2 c 59, where the constants c, Q and p0 are independent of a. I] 4°. The space of coelrains on elementary coverings. Let .1]: (M,(z)) be a family of majorants in C", let U =(U,} be an elementary covering with parameter zero, and let vg0 be an integer. For arbitrary integers «>0 and kgo, we consider the following norm defined on the cochains

of order v on the covering Urns},x

”my: max

sup

sup

lil+|i|§k :u ..... xv mom-wanna.

Di"?

2

M M,(z)

(4-3)

The space of these cochains for which the norm is finite will be denoted by vH;’"‘(U). If dz is some cochain of order v on a wider covering V, we shall say that (13 is an element of the space 'Hf"‘(U), if its norm [4) is finite.

As earlier, we shall set

l

MAI)

:0 outside 12,. In particular, if d: E 0 in (2,,

the cochain d) is equal to zero as an element of the space m°"‘(U). The subspace in 19??" (U) consisting of cochains that are holomorphic in 21, ..., 2,,I in 52,, will be denoted by "m“‘(Ul We set v.9K;"‘(U)= liln'afimwt It is convenient to introduce the notation h—bw

"2fi""(U)=9€;"'-'.

‘11:"(U)=3?;'".

It is clear from formula (2) that for arbitrary or, m, k and v a — l the operator 13 defines a continuous mapping a: '9fi’”>"(U)—>"“.#;’“"‘(U) with norm not exceeding unity, and also the limit mapping a: “Mfume

§ 3‘ l-cohomologies

109

H 15?;"(Ut We denote by “12;“ and if,” the kernels, and by ”W?" and ”‘93:" the images, of these mappings. If agar, and the covering V is inscribed in U, then there is defined also a continuous mapping of the restriction ”fi’""‘(U)—» 'fl'r’fl V), the norm of which does not exceed unity Such mappings. and the corresponding limit mapping, will be called identity mappings. By (L, 0 ”haw. The spaces VJt/fiUa), Ot= 1, 2,

, form a family which we denote by {#4.

The coboundary operator defines the sequence of mappings

cam—LWA—Wm—twavxI—Lvfim—w-n (5.3) This sequence is semi-exact, and therefore we may consider the factorfamilies Ker (6: ".16, —> ”bad/B“— {9a .

These factor-families will be called vllvcohomologies. We shall say that the JI-cohomologies are trivial, if the sequence (5) is exact. 5°. Proposition 2. Given an arbitrary elementary covering U of para-

vgo, the meter zera, given a number 9, 0 ”21% U) is an identity mapping. Hence follows the exactness of the sequence (1) With m:0.

|]

2°. Lemma 2. Let M be an arbitrary family of majorants and let a > 0, v 2 0, and m, 0'”t,)—»W’,/a‘—‘stg,

114

III, Cohomologies ofAnalytic Functions of Bounded Growth

inverse to the mapping 6, is defined. We shall show that the mapping Rv can always be so chosen that its order (see Definition 2 of § 1, Chapter I) is some fixed function a: —» 3(a), not depending on the family of majorants .II. To this end we begin by proving a similar assertion for the mappings

1a,: Kerb):'*‘J€’;—>"+1fl}—>V3f;‘/av“flz, which are inverse to the mappings 6. According to Lemma 1, given an arbitrary family of majorants .ll, there exists a mapping R3, v: 41,0, 1, ..., inverse to 6, and of order equal to the function [imam We shall now suppose that there exists a mapping R2,,_1, inverse to 6, with an order not depending on the family .11. We consider the diagram (7). By Theorem 1 of § 2 and 2 of this section, given arbitrary v; —1, there exists a mapping T": Darla ‘flj’l/Vt’jt", inverse to the mapping 8/62,", with an order independent of .1!. Therefore in diagram (7), we can apply the remark of § 2, Chapter 1, according to which, corresponding to an arbitrary v; — 1, there exists a mapping RI", inverse to B, with an order dependent only on the order of RI“, and T“, and, therefore, not dependent on J]. Thus, we have carried out the induction with respect to m and we can say that there exists a mapping R:, v; 1, inverse to 6, with an order not dependent on .11. Finally, from Proposition 2 of § 3, it follows that the isomorphisms “16“;v, can be determined with the help of mappings whose order does not depend on .11. These mappings (more exactly the mappings associated with them) when suitably compounded with the mapping R; yield the desired mapping R". Clearly, the order of the mapping R" constructed in this fashion, does not depend on the family .11. Our assertion is thus proved.

4°. Cohomologies of analytic cochains in bounded reg‘ons. It is well known that in an arbitrary pseudo-convex open set, the cohomologies of analytic cochains are trivial. We now establish a variant of this classic result by turning our attention to an estimate of the norms of the corresponding operators. Let 0 be an open set in C", and let U be an elementary covering. We consider the norm

||¢||3,u=zsup‘ ump |¢I. ta,,....,,=ll.,n---nv..na. defined on cochains over the covering UnQ. Theorem 2. Let (0 be an open set of diameter not greater than 1, sup— pose 0 < rg 1, let 9 be an r-neighborhood of w, and let U and V be elementary coverings of parameter zero and radii Zr and r. For every holomorphic cochain dz on the covering U n 9 with a finite norm ||¢||g_ u, such that

§4, The theorem on the lriviality of lac-homologies

us

6%= 0, there exists a holomorphic cochm'n \l/ on Va (0, such that a¢=¢ a

lill’llgyéé ll¢lI3,u.

(8-4)

where the constant C is independent afco and r. Proof. For arbitrary integers vgo and kgo we consider the space "[5,

consisting of cochains F of order v, on the covering U n {2, the component s of which are "d-forms of order k (see 7°, §2) with square-summable coefficients:

F= Z I .....z., In

' 11."..11

z§:::::.’tdij.~~'AdinflLoA-"AUW

We define the topology in “L‘k by the norm

, :Ilt,(u..,,_..,,,yl*-

IIFlIn,u= SUP I 2

1mm... hm,”_ "/33. We consider the commutative diagram:

‘

2

0—»1H(U)——»1L°J»‘Ll#w

3].

at

a

0—»°H(U)——.°L°#»°LI J» 0

(9.4)

III

, ”(9)

, L0 A, L1 J,"

III

0

0

0

Here 15‘, k=0, 1, 2, , is the kernel of the mapping 0: °U —> ‘lf‘, that is, the space of ”d-forms in 12 of order k with square-summable coellicients, and ”H( U) is a space of holomorphic cochains of order v on the covering U n!) with the norm ”Winn:

SUP: ”452°.“uxvlliumuwzvy m... V

Hm) is the space of square-summable functions holomorphic in Q. We shall prove the exactness of the rows of this diagram. Let I-‘e-VL'”l and ”1”" =0. We choose 20, .. . , z‘. arbitrarily and we consider the ”d-t'orm

f=f..,, .....= 2 12211::i: dil. A .hvmdk

Adi/k-

116

Ill. Cohornologies of Analytic Functions of Bounded Growth

It has a finite norm N f ”I,” .. z and satisfies the equation ”11F =0. According to the theorem of Harmander, formulated in 7° of § 2. there exists a ”d-form g such that

d£=fl

llgllu in..... x "gnu/Hum,”

(10.4)

Assuming that the form g depends in an anti-symmetric fashion on 20, ..., 2,, we make up the cochain (i=2 gUm A A U“. It follows from (10) that

”dG=F,

llGlln,u§12||F||n,u-

Thus we have established the exactness of all rows of the diagram (9), except the lowest. The exactness of the lowest row is proved in an entirely similar fashion if we remember that the norm ll - “11.11 in the space L" c OL" is equivalent to the norm || . “at We now turn our attention to the columns. Making use of the notation of Lemma 1, we write for every cochain F 5‘15 such that 8F =0,

gz,,...,,.=(V+1)[(h.fz,z.,...,xv)'l12 '5 We note that 0g h,§1 and that for arbitrary fixed 21, .. , , zv the integrand differs from zero only when IZ—z1lé2. Therefore the square of the right side does not exceed

CI lf1,z;,..t,zv|2 ldzdfl. where by |f...|2 we mean 2 [fif- ""”‘|‘, and C depends only on n and v. Hence

I lg... “ml: Id: m: C I If,,

“(01‘ Id: dZ dz dzl.

Therefore the operator

FoG=Zgnumz UnA-uA Una-‘1} is defined and continuous. It follows from the calculations made in Lemma 1 that aG=F. Thus we have established the exactness of all the columns of diagram (9), except the leftmost. The exactness of the leftmost column follows from Theorem 1 of § 2. Chapter I. We? may therefore assert that for an arbitrary cochain 455” H(U) satisfying 845:0. there exists a cochain v115"H(U) such that a t1: = d), and

ill/Illn,u§C||¢||n.u~

(11.4)

We remark that we are justified in supposing that the constant C in this inequality is independent of a) and r. since the constants which appear in the preceding argument have this property (see 5° of § 2. Chapter I).

§4. The theorem on the triviality of I—cohomologies

117

The right side of (ll), clearly, does not exceed the norm C||¢||}',.u. 0n the other hand, every region KW“; au-n tw belongs to Um_____ ,V together with its r-neighborhood. Since the components of the cochain III are holomorphic, it follows from Proposition 1 of §l that we have the inequality

C v2.3?» |~II,.,,...,,J§7 |||lI........z.||z..w,m) where C does not depend on a) and r. Hence "1/413n C “ulna”. Combining this inequality with (11), we arrive at (8). I] 5°. Examples of non-trivial .ll-cohornologios. In this subsection, we shall say something about the general problem: to write down all families of majorants J! which correspond to trivial .lI—cohomologies. Certainly, the families of type J are far from exhausting the set of all families for which JI—cohomologies are trivial. For example, a well-known theorem of Oka-Cartan-Serre provides another example of such families. A more general class of families of majorants of this type has been given by Harmander [2]. We now produce two examples of families of majorants with nontrivial .l-cohomologies. In the first example, the family .1! will be so chosen that the sequence (5.3) is not algebraically exact at the term ‘96,. In the second example, the mapping 6: Mop» bf, is not a homomorphism.

Example ]. Let n= 1, and 1 M..(Z)=(IZI+1)‘6XP(-;IImZI”‘), a=1,z..., where e> 0 is an arbitrary fixed quantity. It is not difficult to see that the sequence of functions M,(z) forms a family of majorants. Suppose that U = { U,} is an elementary covering of parameter zero, and that R + is the open half-plane Im 2 >0, and R _ is the closed half-plane Im z§'0. We consider the following holomorphic first order cochain on the covering U : ¢=Z ¢zmz Uh" U“,

where ¢'hn(z)=

0,

zl,zzeR,;

1,

zleR+,

zzsR_,

[—1,

zlelL,

zzeR+.

zsUn n U";

It is clear that the components of the cochain ¢ differ from zero only in the strip |Im zlgrw) and are bounded in the ensemble; It follows that

118

ll]. Cohornologies of Analytic Functions of Bounded Growth

the function ()5 belongs to the space ‘x’,( U). It is easy to verify that 19¢ =0, and therefore, 436 lfl’lw). We shall show that for any covering V and integer a, there exists no cochain Ille°i(V) such that all =45. It will then follow that the sequence (5.4) is not algebraically exact at the term ‘17,. Let us suppose the contrary. Then suppose that on some covering V= ( V,}, there exists a cochain ¢=Z lb, V,e°.9?;(V) such that Bill=¢. This equation implies that for arbitrary zl, zzeR+ or 2‘, zzeR_ , :11“ so" in V,‘ n V“. Accordingly, the functions ill, with 26R, are restrictions of some holomorphic function (Iii, defined in R1. But the inclusion relation Ille°.}f;(V) implies the in1 equality

|¢1(Z)I§ C(IZ|+1)‘ eX13 (-711m Zl”‘)~ Hence it follows that $1 50, that is '1’ =0. Therefore, the equation 31p: 45 cannot hold.

Example 2. We again suppose n=1 and we let h(a') be a non-negative function in R1 which is not identically zero, which is infinitely differentiable and has a support contained in the segment [‘l,1]. As is known, the Fourier transform of such a function satisfies the inequality

[5(1)|§u(x)¢xp(|yl),

Z=X+iy-

where u(x) is a positive function decreasing at infinity faster than an arbitrary power of be]. We set

and

M.(Z)={

fl.(x)=sup (M05), IX—X’lél—Z'"}

(IZHW—lflAXNXNIYIL lXIélYl, 6>0. m=l,2.---. (IZI+1)‘"#.(X)BXP[|y|+(1+e)|y—XI], lyIEIXL

We shall show that the mapping a: 0X, —> 1x, is not a homomorphism. We assume the contrary: 6 is a homomorphism. Then, in particular, if U={U,) is an elementary covering of parameter zero and radius %, there must exist a covering Valid an integer at and a continuous operator

R1 l931W)-*°:’€”.x(V)/°2'..(V) such that the composition ER is an identity mapping. From the continuity of the operator R, there follows the inequality

infllltfi—Bxllfiv. xesmsc Ii3¢lif.u»¢5°xi(u)-

(12.4)

We shall show that this inequality cannot hold, by constructing a sequence of functions for which the right side of (12) is infinitely small compared to the left.

§4, The theorem on the triviality of l—cohomologies

119

For every integer 1:2 we consider the cochain 1

IP =ltm, U3, A Um the components of which are defined in the following fashion. Let S1 be a closed sphere of radius 1 with center at the point M. Then

_

h(Z)

¢,.,,,(z>=

z—il’

zleS‘,

zzesl,

_z—i}.’

2,651,

2165};

5(2)

_

zeUnnUn.

tame) :0 {zhzzésr

21,1255»

It is clear that for arbitrary z, and z,

III/1h.,(2)|§215(l)|§2/1(X)expflyl). It follows that the cochain 1/11 belongs to the space 1.9mm. We shall a derive a bound for its norm. Since an arbitrary component of it is different from zero only in the i~neighborhood of the point i), we have

M =51,a}: ”nag” w III/man M1(z)

I'll/"1.11

< sup I5(Z)| =i§Ix—izlgs lz—illu’(X)eXP(|yl+(l+e)ly-XI) (13.4)


axll.=|Ԥgr>g*

; Csup(|x|+l)

1

5(2)

z—M‘W’l M42) fix)

1‘ x-A

My;

: c" IIF-l [jfbjl/T] rF-l [x00]

50c)

x—i}.

—x(X)

L:

(14.4)

L;

where F “ is the operator of the inverse Fourier transform. The Phrag— men-Lindelof principle implies that an arbitrary function 1 belonging to the space 9f” that is, satisfying the inequality

lx(2)|§ CM,(Z),

(15-4)

ix(Z)l§C'eXP(|yl).

(164)

also satisfies the inequality

Accordingly, the inverse Fourier transform of the function 1 is an infinitely differentiable function, with support belonging to the segment [—1, 1]. Hence, for an arbitrary function 153?; the right side of (14) is not smaller than —I

F

Mac) [x—il] L21R d

where R+ is the pair of rays Io‘l > 1. The inverse Fourier transform of the

product it“? is the convolution xaxl 1 0 h (0» F _ 1[fi]= ‘ a>0 HA Whom. 6m = {m “a‘

Since the function h(¢7) is non-negative. the convolution —i Ma): 01(0) is also non-negative and —ih(a')*91(n')=be“'. ag ——l for some b>0. Therefore

b ||h(0)*91(6)llnrk.l 2% _ F212 "l.

If we take account of this inequality and use (14), we arrive finally at

inr{II¢*—ax1|:y.xeas:)zie—z “1/27

§S. Cohomologies connected with 9~matrioes

121

forarbitrary 11. Combining this inequality with the inequality (l3). we arrive at a contradiction with the inequality (12L ' The examples 1 and 2 can of course be generalized in various directions. We shall take note of only one such generalisation, and that Without proving it. Let i(¢) be some continuous periodic function in the segment [0,21r]. Then the family of majorants

M.(2)=6Xp “Karen—am Z

a=l,Z...,

(17.4)

defines trivial .ll—cohomologies, if and only if the function i(¢) is trigonometrically convex. This statement illustrates rather clearly the idea of Examples 1 and A that non-trivial JI-cohomologies arise when the family of majorants .1! in a certainjense does not correspond to the inventory of functions in the space Xd=lim 9%,. Thus, in Example 1, the space 3%, consists of the single function that vanishes everywhere. In Example 2, this lack of correspondence consists in the fact that the functions of the space at" satisfy the inequality (16), which improves the inequality (15) in the region |y|;|xl. Our final assertion shows that the JI—cohomologies defined by the family of majorants (17) are trivial if and only if there exists an entire function of first-order growth, whose indicatrix is equal to i (45). These observations lead us to the following condition as necessary in order that the family J! should have trivial Jl-cohomologies: the functions M,K cannot be essentially decreased without changing the inventory of functions of the space Jfffllor, mgre exactlyfi there_ exists no family of majorants .ll’ such that .Wl=.;ffl,while JV}C.#2 and i2¢£fi Probably the sufficient condition for the triviality of .lcohomologies should be close to this necessary condition. § 5. Collomologios connected with ?—mntriees

l°. Formulation of the theorem. Let .4 be a family of majorants and let VffizraflUa)‘ ot=l,2,...}, v=—l,0,l,..., be the corresponding family of spaces. Further, let p: 95% .9" be a 9-matrix. that is, a matrix of size rxs, consisting of polynomials in C". For arbitrary v, a and arbitrary covering U, the multiplication of holomorphic functions on the covering U by the matrix p defines the continuous operator

:12 [m(u;]-_, ['eKmUfl'.

(1.5)

where n is the highest order of the elements of the matrix p. This mapping clearly commutes with the identity mapping of the space “Jaw. In

Ill, Cohomologios of Analytic Functions of Bounded Growth

122

particular, for arbitrary a, the following diagram is commutative:

[Véfi+1(%+1)]‘—’> ['JfiqUHHuJJ' [WWJJ’ -‘Lr [‘Jfi+u(U.+.)]'It follows that the horizontal mappings in this diagram are components of the family mapping

PI [WOT H [9741‘-

(25)

Let 9 be the p-operator constructed in Theorem 1 of §4, Chapter II, with m=n. By [‘atf;(U)]'nKer9 we denote the subspace in [Vatfi(U)]', consisting of oochains (1: such that 9(2) ¢,n"w,y(z)50 for all 20, ..., zv. By [Damn Kerfl we denote the subfamily in the family ["X‘T, consisting of the subspaces [V9€(U,)]'nKer 9, 01:1, 2, It follows from the properties of the p [‘EZ'RT). Here

HP ¢II5R§3§3 “7(1)! ||¢l|5"‘§ C(IZI+1)“II¢||5'R.

Ill. Cohornologies of Analytic Functions of Bounded Growth

126

where the constant C does not depend on Z and R. Therefore, the operation of multiplying functions on C"XK with values in the spaces VEf' R by the matrix p defines a continuous operator p: ”H. -* '11,”. The set of all such operators defines a family mapping p: '17 —>VII. We shall use "Ef'RnKer 9, "ILA Ker 9 to denote the subspace of the respective spaces ['Ef"]’, ['I'IJ’, consisting of the functions ((1, ¢=¢LR such that 945:0, 9¢z"=0 for all 26 C", ReK. We use ”II n Ker 9 to denote the subfamily in ['l'l]‘, consisting of the subspaces “HunKer 9. We use the notation '1'! n Ker p B, ‘3! n Ker 9 and so forth, with a similar meaning,

Lemma 2. The sequence

°HnKerpB—'—»°InKer9—>0

(8.5)

is exact.

Proofi At first we describe the idea of the proof. Every element of the space °Zu n Ker 9 is by definition a function on C">3——'—(u+l)(a+2).

every such point belongs to aft“ together with its e-neighborhood, therefore the function 4) is holomorphic and is bounded in the s-neighborhood of the point 2. According to Corollary 1, §4, Chapter II, the operators 9(2) and 9(2) carry functions that are holomorphic in the s—neighborhood of 2 into functions holomorphic in the e r,-neighborhood ofthat point, and their norms do not exceed (3" r,)‘ l, where rx = C 9" (z, A"),

55. Cohornologies connected with 9-matrioes

127

and JV is an algebraic stratification of C”. It follows that for arbitrary zeeffi the functions x, and (It, are holomorphic in the neighborhood S, of the point z of radius er, and l

1

max SUPIXxlfiupllllzl é-— 5“ |¢|=—x ||¢llf"~ 5. s. e" r, eE-E a r,

(10-5)

Now at every point 260', we construct the sphere S, of radius a r,. According to Proposition 1, §3, we can inscribe in the covering S = (S,,ze C") an elementary covering V={V,) with some parameter p and radius be“, where the quantities p, b and q are independent of 3. Suppose that some sphere V, intersects the ellipsoid 2512. Since the covering V is inscribed in the covering S, this sphere lies inside some sphere S,“ The radius of the latter is equal to e r,.§e§ p(ef'+’§, C effl), hence the poinm z and z' belong to effi. It follows that the covering Vn eff; is inscribed in the covering {Suzeef'fl}, and the radii of the sphere V,, intersecting with effi, are not less than

a ‘1' be‘(|Z|+2) ”2_b, (IZI+1)’ where b’=2"’b, and q’=max(q, p). Since R‘§%, we can find a sufficiently large constant Q, independent of Z and R, such that

’7’ (—8 )3 (—R )Q~ IZI +1

_

IZI +1

But this inequality implies that the covering Ug'kn eff; is inscribed in Va eff; and afartiori the covering (15' R n 2f ‘, where ar’ = max (on + 2, Q), is inscribed in Va eff; , and, therefore, in {S,, zeeffl}. Accordingly, we have defined the restrictions x’, l//' of the cochains Z x, S, and Z 4/, S, on the covering Ufn“ n e5“. Eq. (9) implies that

(11.5)

6¢=1’+p it

We set [=2x, U, and i//’=Z up; 1],, where (U,}=Uf'“ne,z:“. If some region U, of this covering lies within the sphere S,. , then the corresponding sphere of the covering Ug’“ also lies within S, and, therefore,

R

, —— “tailJ

Q

,

whe“oe

R

9

r,.g(——).

sinoee§1.So

2+1 9

|Z|+1

,

lllév RI ) eSUP|9(Z)¢|suplx’zlésup 3,. 3,, U.

(125)

Ill. Cohomologies of Analytic Functions of Bounded Growth

128

Combining this inequality with (10), we obtain

IIx'II§r*§C’('—2'R+1—) ‘1' ”we“. We derive the inequality

(13.5)

Z I Q' nwuf-vxgc" 0—1;) Il¢llf"‘-

in a similar fashion We must emphasize the fact that the constants C’, C" and Q’ are independent of Z and R. We now prove the lemma. Let ¢’ be an arbitrary element of the space ‘Zf'kn Ker 9. This can be written in the form 13¢, ¢EE:f‘R, where ||¢||,Z-“ = |l¢’||f'k. If we apply Eq. (11) to the function 43 we obtain ¢_= pill', since by hypothesis 9 (#50. Thus‘ we have constructed the continuous

operator

L1, ,1: 0522, R n Ker 9 a (15%» We °Ef1 R

such that the composition (IL; 1: is an identity mapping. It follows from the inequality (13) that the norm of this operator does not exceed

,, |2|+10’ C( R ) Therefore, according to Proposition 1, the set of these operators defines a continuous operator L”: ”9!, n Ker9 4°11, , such that the composition p L" is an identity operator. It follows from the equations Pp L‘: p (‘L'=0 that the image of L‘ belongs to the kernel of the operator p60. that is. the subspace ”11,41 Ker p 6. Since the operator L“ was constructed for an arbitrary integer ix, the exactness of the sequence (8) has been proved. [I 5". Lemma3. Fur arbitrary v; —I the sequence

'H—”> 'HnKerQ —r0

(14.5)

is exact.

The proof will be carried through by induction on the cohomological dimension 6(Kerp) of the 9-module Ker {p1 .f‘ai‘} 5. We set 6: 6(Ker p) and we assume that the lemma has been proved for all matrices q with 5(Kerq) 0 0

0—»

0—»“H0Ker96—fl>°aer9 L“, 0

if “H

(18.5)

T

p)

L‘WnKerpaitlaerpaO q‘II

.0]

DH

L‘z

—>0,

By establishing the exactness of the left column, we establish our lemma for the case v: —1. Lemma] implies that the second and fourth rows are exact We shall prove the exactness of the third row. The mapping a“ in this row coincides with the mapping (3‘1 of the sequence (7) and is therefore a homomorphism. The kernel of the mapping 6° coincides with the subfamily °Z. Therefore the algebraic exactness of the third row, at the term °H nKerpé follows from the exactness of (7). The exactness of (7) also implies that the mapping (3° is an epimorphism. It remains to show that this mapping is a homomorphism. To this end we consider another commutative diagram: 0—»0H/oziv‘z

l

l

DHr‘iKerp Bflzllaerp

l

0

All the mappings in this diagram, with the possible exception of the mapping 5° are homomorphisms. If we apply Statement B) of Lemma 1, §2, Chapter I, we see that the mapping 5° is also a homomorphism. Thus,ctlhe exactness of the third row of diagram (18) has been completely prove . Let us consider the columns of diagram (18). The exactness of the third column follows from the properties ofthe left column ofdiagram (16). The exactness of the second column at the term “ZnKerQ follows from Lemma 2, and its exactness at the term °17n Ker p 3 follows from the exactness of the sequence (17) with j =0. Thus, we again have the right to apply Theorem 1, §2, Chapter I, from which follows the exactness of the first column. With this, we have proved the exactness of (14) with v: —-1.

95. Cohomologics connected with farms-ices

131

Now, we establish the exactness of the sequence (14) for arbitrary v; 0. The exactness of (14) with v= —l implies that there exists a con-

tinuous operator

B: "Haer9—>“H./“fl,nKerp,

agl,

such that the composition 1:8: “111 n Kerga ‘lflawnKerQ is an identity mapping. In accordance with Proposition 1, the operator B generates a series of operators B“: Ef'Rn Ker 9 —» Ef"/Ef"n Kerp

such that the composition p81R is an identity mapping for arbitrary Z and R, and |Z|+1 a_1

”Bun s c (T) .

(19.5)

We arbitrarily choose vgo, a, Z and R. Let the integer b be so chosen so that b -o 2 §3vz(vz+b). Then

R

U+b

(m)

§§P( ‘35,,Cef'k),

since R1§%. Therefore, if the sphere V, of the covering Ufl: intersects

the ellipsoid e15, the sphere 2 V, lies within ef'“. Since [7:2, we have V,c% (1,, where U, is a sphere of the covering Uf’“. Let 2.), ..., zv be an arbitrary point such that the intersection Vnnu-tn ,1}, is not empty. From what we have said earlier, we can derive the inclusion

14mm14mm:Koceiwceiwznocvmnu-nU..ne:-', where

r=r(Uz’ R): (__R—).+b.

'+"

Izl +1

Let (15:2 (pm m,” U,“ A A Uh be an arbitrary element of the space "Ef'RnKer 9. The restriction of the function thaw,“ on the sphere e?" ' is an element of the space Bf“ 'n Ker 9. If we apply to this element the operator 8,0,“ we obtain a coset belonging to E:°-'/E;°"nKer p. Let It“. 2., be an arbitrary element of this eoset. From (19) we have

inf{Sllp’lI/I.o,.i.,x.—xl, zeEzov'nKerp} "°

§C(£a|r—+1—)a—l

sup , , l¢,.,, 2,0,

”I.

(20.5)

132

Ill. Cohoinologies ol' Analytic Frmctions of Bounded Growth

Restricting each ofthe functions #1o ,v to the region Vun- - -n Kvnef'flfi, we construct the cochain '11:: Wow...» Vw/x A V,” on the covering (L1,;fnef'n. (We can, of course, suppose that'the function 1/110,,,,, ,v is skew-symmetric in its indioes.) By the properties of the operator ol,

(21 -5)

P W = 4’ The inequality (20) implies that

+1 H mi". (22.5) inflllllI-xllffi, ze'EvtnKerp}; 032,13 (%) We remark that on the left side of this inequality, we have the norm of the cochain i/t as an element of the factor-space "Effi/TffinKerp. We estimate the second factor on the right side. Since Izo—Zlé 1, we have |z°|+1§2(|Z|+1). Using this in (22) and substituting the expressron for r, we arrive at the inequality Z +1 " *Iwuft‘igc, (HT) uni“.

Hence, in view of Proposition 1, it follows that the series of operators 142,31 {13 —v v11 defines a continuous operator

A": ”H,nKer9—>“H,.,

oz’=az+b+q,

and it follows from (21) that the composition p A" is an identity mapping. With this the exactness of the sequence (14) is proved for arbitrary v. I] 6°. Corollaries of lemma3. We mention two corollaries which we shall use in the next chapter.

Corollary 1. Let r(z) be a power function on some algebraic stratification, not exceeding the function I", which appears in Corollary 1, §4, Chapter II. Then for arbitrary Z and R and ¢, 4) being holomorphic and bounded in Zea“. we have the equation ¢=¢g+pi//, where 459 and up are fimetions holomorphie in ez' R, and

3’33 I¢gl g C ( IZIRH ), sup {r(z)m 2'13)“,s 5) ¢(z)l, ze2ez' ", r(z) (23 5) R

q

g“ (m + 1 )}

'

for some constants C, e and q, independent of Z and R. Proof. We consider diagram (16) as a whole. In view of Lemma 3, all of its columns, beginning with the second, are exact. It follows from

§5. Cohomologies connected with 9-matriees

133

Lemma! that all of the rows are exact, beginning with the third. It is clear that the second row is algebraically exact, and the mapping i is a homomorphism. Applying Theorem I, §2, Chapter I, we can establish the exactness of the left column. We consider a new commutative diagram:

0

0

°z—>°HnKer96/°HnKer9—> 0 0

i

z———>"Ilr'\Ker96‘>a

0”

fi

I

‘aer9—>0

1Z

0

The exactness of the third column of this diagram follows from the exactness of the left column of diagram (16). The exactness of the second column follows from Lemma 3. The exactness of the third row follows from Lemma 1. We consider the second row. Its algebraic exactness at the term °17 Ker 9 6 is obvious. The mapping a in the second row is an epimorphism since by Lemma 1, 61°11 n Ker 9 a)= 1!! n Ker 9. The mapping 6 is a homomorphism since the mapping 6° in the sequence (7) is a homomorphism. Thus, we have established the exactness of the second row. Applying Theorem 1, §2, Chapter I, we conclude that the first row is also exact. From the exactness of the first row, it follows that

for arbitrary or there is a continuous operator I: ”17, 09 B/“Hun Ker 9 —§ °.”2’,/°fl',n Ker 9,

fi=fi(u),

inverse to the identity mapping. Arguments like those of Proposition! show that the operator I generates a series of operators

11.16 0155'" n Ker 9 B/"Ef‘ R n Ker 9 —» "£2";- "l‘lyf' R n Ker 9,

(24.5)

whose norms are bounded by functions of the type |Z|+ l )‘1 . c (Tr

Let dz be an arbitrary function belonging to 1511' R. We apply to it the arguments of Lemma 2. From the decomposition (11), we find 61’: —p¢9Il/‘. Therefore the cochain 1’ belongs to the space °Efz R.” Kerga. Now in (24), we replace a: by a’ and we consider the cocham 1’ as an

Ill. Cohomologies of Analytic Functions of Bounded Growth

134

fi’=p(u’) be element of the factor-space in the left side. Let ¢’e°.9!,zu‘, From the a cochain running over the image 1’ under the. mapping 1;, K: bound on the norm of the operator 1,. R we derive the inequality

2 +1 . Hz, ”3'“. infudz'llfikéc (%—)

(25.5)

The properties of the operator 12.x imply the equation (26.5)

9(6¢— ¢’)=0.

It follows from Eq. (25) that for arbitrary e>0, we can find a cochain (#611. n x’ such that Z +1 ‘1

(27.5)

Il¢'||§:‘§c (%) 1|x'||,f.-*+e.

If the first term on the right side is different from zero, we can suppose that s is equal to it. Then we obtain the inequality

(28.5)

[Ida'llfizkézc (%) "rue“. 11

Let us suppose that the first term on the right side of (27) is equal to zero. Then from (11), we have 6¢=pi//, and, accordingly, 9(2) ¢(z)EO in ef: “A Then we may set (#50, which again leads to (26) and (28). We denote by ¢geE§:" a function such that ¢'=6¢g. We find a bound for llx’llf: R with the aid of the inequality (12). We remark that in the construction of the decomposition, we can replace the function rz by r(z), since by hypothesis r(z) has those properties of r, which we used in 4°. Therefore, we can derive from (12) the inequality

z|+1 0 llxll. _c(—R ) 7

2:11:

xsu

|

R

0

z — . P{r(2)mséuzl:n| 9 (2, we» , ze 421-“, 1 r(z)—(IZ|+1)}

Together with (28) this inequality yields (23). It follows from (26) that

¢—¢geEf¢“nKer9. Therefore, in View of Lemma 3, ¢—¢9=pw, where WeEfn".

|]

Corollary 2. Let V be an elementary covering of parameter zero, and let dz be a cochain of order zero on this covering, and let it be halomorphic and bounded on some open sphere S, and let 9 645:0. Then on the sphere is there exists a bounded holamarphicfunction ill such that 9(45 — 61h)=0.

§5. Cohomologies connected with 9-matriees

135

Proof. The cochain 4) belongs by hypothesis to the space °E{-’n Ker9 B, where z is the center of the sphere S, and Zr is its radius. Let i/I’e°9,’f(3f be an arbitrary element of the coset 1,, , 45. It is easy to see that the function the 1‘11,” such that add = 11/ is the function we are looking for. l]

7°. Lemma 4. For an arbitrary family of majorants J! and for arbitrary integer v go the sequence (29.5) [56,} —"> [96,} n Ker 9 —» 0. is exact. The proof of this lemma is similar to the reasoning of 6". We arbitrarily choose the integers v20 and at We set U1: {U,} and U,” = {K}. Let the points 20, , 2,, be such that the intersection V,» n n Vang“,2 is not empty. Since 1 s '(U.+2)§TT(UJ=T‘,

we have the inclusions

V...n---nKvnnwcKoce:°"cer"=2V..,cv,.,n-~none“ r=r(U,,+2). Let 4: be an arbitrary cochain in the space “afi([L)nKer9 and let on“ 2., be the component of q) corresponding to the regions Um, ..., Uh. We apply to its restriction on 4°" the operator 8%,, constructed in subsection 5°. Let Wm M,» be a element of the onset 3,0,, on, ,v. From Eq. (19), we have

m...... .v—xl. 16E:°"}§C(|2o|+1)‘"f:gI l¢.o....,z.l. inr{sup . e‘a,’ Hence, sinoe r.$”“. Proof. We fix an arbitrary point 26 C" and we set 9 = 9(2), {4 = 9(2), and so on. Let ¢ be an arbitrary element of the space .9". From the equation ¢_91¢=Pig1¢=92(¢‘91¢)+P2g1(¢—91¢) “-1) it is clear that the series 92 ((15—91 ()5) belongs to the arithmetic sum of the subspaces Pi 3m+l72 5””:(PIGPz) yaw“, and accordingly 92(¢‘91¢)=(P1$P2)g12 92(¢—91 ¢)

=P1 7‘1 912 92(‘l1‘914’)‘i'llz 715: gt: 92(¢—91 ‘1”We substitute this expression in the right side of (4) and we move the term 1111!; 9,2 92(41—91 45) to the left. As a result. we obtain the equation

ages—91 ¢—p. m 9” glob—91¢) =P1gi¢—Pi7‘tgu 92(¢_91¢)

(5-1)

=P2 7‘2 gr: 92(¢—91¢)+P2 gz(¢_91¢)a from which it follows that the series 41’ belongs to the intersection pl 5”“ n p; 5’". Since the ?—module 5’ is flat, it follows that p Y5= p9” ® 5’ for an arbitrary 9-matrix p (see 2°, §3, Chapter I). Therefore Eq. (3) and Proposition 4, §3, Chapterl imply that the intersection p, 5”" ('t Y“ coincides with p9”. Hence 9 ¢’=0. Therefore, applying the operator 9 to the left side OHS), we obtain the relation

9¢=(9—9p, n1 9” 9;)91 ¢+(9 pl 1:, 912).”); (it, which is the desired decomposition.

I]

Corollary 1. Let the f—matrices p, p0, . .. , p, be such that p9‘=po?‘°n---np,9".

Then the p-operator 9 admits the following decomposition:

9:2/1191,

(6.1)

where 9‘, i=0, i.., l, is a pA-operator, and AA is an operator in 5”, having the following property: for an arbitrary 26C" and 6, 0 n— m, and is normally placed in C". Then there exists in the space CW a proper algebraic variety [,1 such that for an arbitrary point Z 6 C”\(C,,xu) we have the equation

p(Z).S”‘nrlm19"=rl,,,p(Z).V‘.

(20.1)

Praafi The preceding theorem implies the existence of a subset .1! c C of first category, such that for an arbitrary point will the submodule pw=pw 9.,“ of 9; is unmixed and of dimension m—hx and normally placed in CV. By Propositioné it follows that for arbitrary ), Thus Zme C1 the condition (Zn-Zn) fepw implies fepw (m>h=n-d

IV. The Fundamental Theorem

150

we have established the equation

Pwfl(Zm-Z..)91§=(Zm—ZM)PW.

Wél-

(211)

We imbed the space CI in the space C”“=C"XC‘, the points of which will be denoted by (2,2,). Any polynomial in C” can be looked on as a polynomial in 0'“, which is constant in Z". We. thus obtain a mapping of the ring 9:91 into the ring yz-Zz... consrsting of polynomials in the variables 2, Z,,,. We denote by [7 the W-matnx (2”, —Z,,,) p, and we denote by e the unit matrix of order t. Applying Theorem 1, §4, Chapter II to the W—matrices p,i;(z,,—Z,,,) e, p®(z,,—Z,,,) e, we construct the operators

rig; r1, 2; d°,g°; d’.g',

{2-1)

which act from the space 3" to the space 3’", where t’ is equal to t, s, t+s, and which have the following properties: there exists an algebraic stratification JV = {NV} of the space 0'“, such that on every set NV\Nv+l the coefficients of all these operators are rational, regular functions (in fact, Theorem 1, §4, Chapter II guarantees the existence of such a

stratification for each of the matrices enumerated above-in order to construct the common stratification, it is sufficient to apply Proposition 3, §3, Chapter II)

From (21) and Proposition 4 we derive the equation

1(z,Zm)=/1(z, Z...)d(z, Z..)+/1°(z,z,,)d°(z,z..),

wéll, (23.1)

where A and 1" are polynomials in the operators (22), the matrix 11, (zm—Zm) e and the projection operator 9”" 45/". It follows that the coefficients of the operators A and A" are rational, regular functions on each of the sets NAN, H. We fix an arbitrary index v20 and we resolve the variety NV into irreducible components. Let N' be one of its components not belonging to NV“. We divide the coordinates in the space C"+1 into two groups i=0), 2,") and w and we apply Proposition 5 to the variety N’. We assume that for this variety we have case II. Then the variety N’, with the exception of its discriminant subvariety N5, is projected into an open subset in CW, and accordingly into an open subset in C”. Therefore the set Cw\./( is everywhere dense in the projection of N’\N,, on Cw. Therefore, the set C”+‘\(Cux.lt‘) is everywhere dense in N’\N,,. We remark that the subvariety Na, together with the subvariety N’ n Nu“, is nowhere dense in N’. This follows from Lemma 1, since the variety N’ is irreducible. Hence it follows that the set C"+1\(C,,X.IIXC‘) is everywhere dense in N’\Ivj,. Since Eq. (23) holds on this set, and both sides of the equation

§2. Local p-operators

15]

have as coefficients the rational functions in N’\N,,,,,, and these are non— mfinite, this equation is satisfied on the whole set N’\N,,H. We denote by u, the projection on CW ofthe union of all the irreducible components of the variety NV, for which case I of Proposition 5 holds. From what we have proved, Eq. (23) is valid for all points wéu= way. We show that the variety u satisfies the conditions of the theorem. In fact, the left side of (21) always contains the righthand side. Let wéu. Since the operators d and 11° vanish on the lefthand side of (21), the operator I also vanishes on the left side in view of (23). By the properties of the operator 1, this means that the left side lies within the right. [I § 2. Local p-operators

From here on, until the end of the chapter, we fix an arbitrary 9matrix 11: W 457'. We recall that the ring 9 is interpreted as a ring of polynomials with complex coefficients defined in C". The points of the space C" will be denoted by z=(z,, ...,z,,) or 6:05,, ..., C"). The letter 2 will often be used to denote a fixed point, at which some p-decomposition is carried out. The letter C will always denote an active variable—active

in the operators that occur in these decompositions. The content of this section consists in the following: beginning with a p—decomposition, corresponding to some value of the parameter m, we construct a p-deoomposition, corresponding to a larger value of m, having some local property, which we shall discuss later. 1°. The initial p—decomposition. We shall now partially reproduce the fundamental theorem of Chapter II. We fix an arbitrary integer m, lying between 0 and n—1. We set n=(.fl, ..., C"); we shall also regard r] as a point in C", lying in the corresponding coordinate subspace. According to the fundamental theorem, as applied to the matrix p and the variables 7;, the identity operator E, in the space 5"[11] for arbitrary 26 C" admits the following decomposition:

E..=d“(2)+p(z+n)g"(z),

(1.2)

where the p-operators d'" and g’", in particular, have the following properties: 1. The operator d”'(z) vanishes on the subspace p(z) 3’1"].

2. The coefficients d{=6,d"'n’ and g{=6.g‘“q’ of these operators differ from zero only if

“balm-1.

l= ~L(K e020,

where K is a constant, and L: 2’" a Z'" is the linear operator constructed in Chapter II.

152

IV The Fundamental Theorem

1 We have the inequality

maxtldll, Igz'nga'Lm-WH'I“, where «:1 is some power function on a certain algebraic stratification M" of the space C". As a consequence of this property, we have 4. For arbitrary e, 0 .9” are arbitrary operators. Their tensor product is defined as:

A ‘3 3 =2 ’7‘ 5445") ® Z "*iw‘“) = Z '1‘ MAM") ® 81-03”). The operator so obtained acts from 9’“ to 5’”. Let us dwell for a moment on a particular case of tensor multiplica— tion of operators. First of all, we note that every series (1550, 0:0[n* can be looked on as an element of the space .9’ [1]], depending analytically on. the parameters a) in some neighborhood of zero; in this case we write ¢=¢(w). We further consider the operator

5f®Em= 2 «MM: yeah]. iell‘

§2. Local p-operators

153

This operator acts on the series 4360 by the formula

(powwow).

(3.2)

Let A': 5”[n] —n9’ be an arbitrary operator which carries the space 0D,] into 0. We write it in the form Zn‘A{(w} 6], where A{(m)e(9[w] and we assume in addition that all the series A{(w) converge in some common neighborhood of zero. In this case, the sum 2 n‘A{(w)6] can be regarded as an operator in 5/[71], which depends analytically on a: in this neighborhood; to accord with this, we write the operator A as Aha). Then we can derive from (3) the following important formula

(A ®Ewl¢=A(w) (M10),

#560-

Let us make one further remark. Let A be an operator in .9[n]. Admitting to a certain lack of precision in the notation1 we shall sometimes regard it as an operator in .9’. Then its action will consist in eliminating from the series the? those terms containing a), and then applying the operator A. The resulting series belonging to 5” [n] will be regarded as an element of the wider space 5’. 3°. Formulation of the theorem. We denote by 033’" the space in 3’", consisting of series of the form p 2 mi (hi, 1

where all the they", and we write wi=émw i=1, ..., [1. We denote by p(z) w .5" the image of the mapping p(z): (1)5" 4.5". Theorem. Suppose that the equation p(z) 3"nw.9"=p(z) (119’, is satisfied at a given point 25 C", Then the identity operator in 9’ admits the decomposition

in which

E =9‘(z)+p(z +tl“) 54* (Z), 9*(Z)=d*(2)® Em.

9* (Z)=g*(1)®Em

(4-2) (5-2)

and d* (z) and g‘ (z) are operators acting from y‘ [n] to .9" and 5”, respectively, while

I. The operator 9‘12) vanishes on the subspace p(z) .9“. II. For arbitrary e, 00, the closed sphere in C" with radius e and center at the origin. Theorem. If the submodule p9‘c9I is primary, then the p-operatar 9=d" satisfies the following inequality:

6" rllisupl19(l.€)¢(z)l. Iélée“ r(2)} ésuplld(1,D)¢(Z)l. ZEN0(Z+ 11.). P(z, LEE“ r(Z)}~ (

23

)

The point Z 6 C", the number a, 0(|9“(Z,§)tl5(Z)l. lfiléa“ 13(2)) ésup lld‘(z, D) ¢(Z)|, zeN‘h(Z+ (1.), MI. was“ ri(Z)}

3 .

( 13)

governing the functions 4), which are holomorphic in Z + [4. Here “(2) [S a power function on some stratification N‘. We write q =max q» and we denote by JV the product of all the stratificat ions .4" and .4”. We choose the power function r on f in such a way that r§min r‘. We

§4. Noetherian operators

175

substitute 6=e'r(Z) in (30) and we combine this inequality with the inequalities till). We are then led to the inequality (29), in which r is a power function on the stratification .M I]

§ 4. Noetherian operators The construction introduced in §3 for the Noetherian operators admits arbitrariness. For instance, a change of coordinate system in C" leads to other Noetherian operators. It is clear that there are also other methods for constructing the operators d(z, D), satisfying the theorem of §3. We now describe some of the classes of differential operators having this property.

1°. Noetherian operators in the wide sense of the word. We fix a 9-matrix p: 9—. 9'. Definition 1. We shall suppose that the submodule pfcg' is primary. The matrix 6(2, D) of size Ext will be called the Noetherian operator (in the wide sense) associated with the matrix p, if I. This matrix consists of dill‘erential operators with polymomial coefficients. II. There exists a proper algebraic subvariety N,‘ of the variety N = N(p) such that for an arbitrary point 2 EN\N,, the condition

3(2, D) ¢(Z)|u=0 is necessary and sufficient in order that the function 4150} belong to

p (9%The manifold N, will be called exceptional. It follows from the theorem of §3 that the operator d(z, D), constructed in 1", is a Noetherian operator in the wide sense for an arbitrary choice of the coordinate system in C", if the submodule p9" is normally placed and N*=®. In fact, the theorem of § 3 remains valid when the operator d(z, D) is replaced by an arbitrary Noetherian operator in the wide sense“. We shall establish a partial result, which is sufficient for

our purposes.

Theorem. Let pray be a primary submodule of dimension n—h, normally placed in the coordinate system z=(v,w). where v=(z,, ...,z,,) and w=(z,,+,, , 2,). Further, let 0(2, D) be a Noetherian operator in the wide sense, containing diflerentiations with respect to the variable v only. Then the theorem of § 3 remains valid when the operator d(z, D) is replaced by the operator 6(2, D). 8 See V. P. Palamodov [14].

176

IV. The Fundamental Theorem

Proof. We shall show that there exists a matrix A, consisting of

rational functions, such that

(1.4)

d(z, D)=A(z)3(z, D).

In every element of the matrix 6(2, D) we replace the operators (1/1"!)D; by the functionals (5?, r]=(§1, ..., 6,). We obtain the matrix 6(2, 6), in which the rows are the linear functionals 13,12, 6): .9" [7]]a C, k =1, .. . , E. Let y be the highest order of these functionals. We consider the matrix

(042,6)77”,1gk§E,]i|§y,1§r§t)

(2.4)

ofsize Ext Zr Let M be a minor ofthis matrix ofmaximal order p, and not identically singxlar. Since the matrix (2) consists of polynomials, the minor M is not singular on the set N\N’, where N’ is some proper subvariety of N. Let a,=(1,, i,)etZ”+, «=1, m, p be the indices of the columns, and 1, ..., p the indices of the rows making up the minor M. We consider the system of linear equations $1,406.42,6)r]“'=d(z,6)i1‘°,

o=l,...,p,

(3.4)

in the unknown vectors 1,42) of order e (2 is the number of components of the vector 11(2, 6) n“). This system has a solution consisting of rational functions, in which the denominators are det M. We write 1,,(2)EO for k=p+1, ...,E. We shall show that 15 21:1. (2) 0, (z, 6) = d (z, 5) . (4.4)

on the set N\N”, where N"=N'UN* UN,” and Na is the discriminant subvariety of N relative to y. We fix an arbitrary point Z eN\N". In the neighborhood of Z the variety N is given by the equation v=u(w), where v(w) is a holomorphic function. Since the operator 6(2, D) contains differentiations only with respect to 1:, we have in the neighborhood of Z 3(2) D) (11— ”(M)"i In = 0(2, 6) 11'",

(I, i)et 2"“

where (v—v(w))"‘=(v—v(w))‘e,, and e, is the column with index I of the unit matrix of order t. It is clear from the construction of the operator d(z, D) that it contains dill'erentiations only with respect to v, and therefore, has a similar property. We now fix an arbitrary couple (1;!) and we consider the system of equations

Zua(z)6.(z.6)n"=a.(z,6)q*vi,

k=l....,p,

§4. Noetherian operators

177

With respect to the functions u,(w), which are defined in the neighborhood of W,.where Z =(V, W). This system is soluble, and its solution is holomorphic in the neighborhood of W, since its matrix is the minor M which is non-singular on N\N’. Therefore,

5:;(1, 5)(n"'-Zua(W) fl“)=0

(5-4)

for all k=1, ..., p, and also for k=p+ 1, ..., E, since the rows 6,,(2, 6) With k=1, ..., p form a basis for all the rows of the matrix (7(2, 6). Hence

6(2, D)f(2) l~=0, where

f(Z)=(v— ”(“9)"i — 2 MW (v— "(M)“? 0'24 Since 6(2, D) is a Noetherian operator in the wide sense, it follows that fE p03. Therefore d(z, D) f(Z)l~=0. Since this equation is satisfied in particular at the point Z, we have

(HZ, 6) (IN—Z 14.(W)rl")=0. Combining this equation with (3) and (5), we obtain 5

r

[21,42n 6)—d(Z, 6)] n"‘=0. The couple (I, i) being arbitrary, the quantity in square brackets is equal to zero. Since the point Z EN\N" is arbitrary, Eq. (4) follows. Eq.(4) implies Eq.(l), in which A(z) is a matrix consisting of rational functions which are regular on N\N”. In the theorem of §3, we now choose the variety L so that N" c L. Proposition 2, § 3, Chapter 11, implies that for arbitrary ZEN and e, 00. Therefore, one of

the operators 3,, has the form a

a +a

‘ 621

6

2 622 ‘

where either al or 1/12 is different from zero. But in this case 6 6 (“1 5—11+a26—zz) (zz—z, z;)=a2—a, 23. The function 412—011 23 is obviously not identically zero on N, and this contradicts the hypothesis that d(D) is Noetherian. The contradiction that we have obtained proves that the matrix p has no Noetherian operator with constant coefficients. It is very easy, however, to construct a Noetherian operator with variable coefficients for this matrix. For example, the operator

1 a,o= (2)

6 6+ 62, SE;

54‘ Noelhen‘an operators

185

is Noetherian. We shall prove this. Since the result of applying the operator to the polynomials z}, 2% and 22—2, 23 is equal to zero on N, the operator vanishes on p‘d’za for arbitrary ZeN. Conversely, suppose it is known that

5(2, D)¢(Z)lu=0,

4550,.

(17.4)

We shall show that the function ()5 belongs to pfl’z’. To this end we expand the function in the neighborhood of the point Z =(0, 0, Z3)eN in a series of powers of 21 and z; ¢(z)=z;¢u(13)z'iz§

I-J Here ¢_.,.(z,) is a function that is holomorphic in the neighborhood of 23 . From (17) we have the equations whence

¢00(ZJ)EO’

¢1o(23)+23 ¢m(13)50y

¢(Z)=(Zz—Zi 23) ¢01(23)+‘ Zzzd’tflzflz'i 2J2l+1.

The second term can be represented in the form

2} ¢1(Z)+Zi z2 ll’z(z)+z% W39),

where 1/1,, W: and [/13 are functions holomorphic in the neighborhood of Z. Therefore, we have established that the function ()5 belongs to the space p93®02=p0§. Consequently, the condition (17) is, in fact, necessary and sufficient, in order that (pep ‘6’}. We still must show that the submodule 119’ is primary. We note that the line of reasoning given above shows also that the condition (17), when applied to the polynomials ¢EQ is necessary and sufficient in order that ([261; f3. We now find the radical of the ideal 179’. If the polynomial f vanishes on the zg—axis, then f’ gep 5?’ for an arbitrary polynomial gem Conversely, if the polynomial f belongs to r(p.?3), then some power of it belongs to p 9". Accordingly, it vanishes on the za-axis. Thus the ideal r(p 93) consists of polynomials vanishing on the za-axis. Suppose the product fg belongs to p93, and that the polynomial f does not vanish identically on the zg-axis. Then the equation 6(2, D) fg hv=0 implies that 0(2, D) g IN=0, whence gap 93. Therefore, the ideal p .93 is in fact primary. 5". Noetherian operators associated with an arbitrary finite P-module

Definition 2. Let M be a finite 9-module. We choose a representation in the form

MEW/p.

(18-4)

186

IV. The Fundamental Theorem

The set of Noetherian operators associated with M is defined as the collection of Noetherian operators associated with the submodule p c 9‘. The set of Noetherian operators associated with M depends, of course, on the choice of the representation (18) (and also on the choice of the primary decomposition of the submodule p). Therefore, in speaking of this set, we shall indicate the corresponding representation of the module M. In the next section, we show that the choice of the representation ( 18) does not essentially influence the form of the normal Noetheriau operators associated with M. We remark that Proposition 1, §1, implies that the set of varieties N‘, associated with the submodule p, does not

depend on the representation (18), since the module M itself characterizes them.

Proposition 5. Let z and w be two groups of complex variables. Suppose, moreover, that

Mag/119:,

Lax/«19:

are arbitrary 9,- and 9w-modules, and {N‘c C", 6‘(z, D), i=0, .. . , I} and {K“c C', g“(w, D) u=0, ..., m) are the collections of varieties and Noetherian operators associated with these modules. Regarding the matrices p and q as 9-matrices, where 9:9,”, we construct the fi-modules .l = 97p 9’ and 3’ =9‘/q 9'. Then

Tom-ll. 2’)=0, ig1, and {N‘xKI‘c 0'“, 6‘®g“, i=0, ..., l; u=0, ..., m} is the set of varieties and Noetherian operators associated with the module .1!@3’, represented in theform l®$g§“/[p 9‘"+q9"] (see Proposition 3, §3, Chapter I). If the operators 6‘ and g“ are normal, then the operators 8"®g‘l are also normal. We leave the proof of this theorem to the reader.

§5. The fundamental theorem 1°. Holomorphic p—l‘unctions Definition. Let p: 9"»? be a y—matn'x, and let

p9‘=vm~--nm be a reduced primary decomposition of the submodu le pg'cgv, and let d: {d (z, D), i=0, ..., I} be the associated set of normal Noethen'an

§ 5. The fundamental theorem

187

operators. Further let 0 be an open set in C". A p-function holomorphic 1n 0 is defined as an arbitrary set of functions f = (fl. i=0, ... I}, defined on the sets N‘ of), where N‘=N(pA), and satisfying the following condition: For an arbitrary point 2 we can find a holomorphic function eo'z, such that in the neighborhood of Z

f‘(2)=d“(z, D) Fz(z)|m,

i=0, ..., I.

The functions f‘, i=0, ..., I, will be called the compon of the p~function f. Let .ll: {Mam} be a family of majorants in C" and (2,, a=l, 2, ..., open sets in which the functions M, are finite. For every a we consider the normed space 9?; (p, 11), consisting of p-functions holomorphic in 9,, for which the norm:

Ilf||.=m;1x 5:9 lf‘(1)l M.(Z) ' is finite. If alga, then the identity mapping Jfi{p,d)—>3fi.(p,d}, is defined and continuous. Therefore, the spaces 3?; (p, d), «=1, 2, .. . , form a family of spaces. This family will be denoted by 391415743}, We shall sometimes abbreviate, writing afflpl and 39,97} instead of 9?;(p,d} and 361(1), d}.

2°. lsomorphic expressions of the family 31"“), d). We recall that in §3, Chapter III, we considered the family '16,, v=0, l, For arbitrary v20, “of, is the family of spaces 972W), a=l, 2, ..., where UK is the elementary covering of parameter zero and radius ea. and Kamila.) is the space of holomorphic cochains of order v on the covering Uu = { (1,}, for which the norm

||¢||au.= sup sup{””;w+'v'.zev..n-~nv..nn.} [Du-y‘v

‘1

is finite. Let 9 be a p—operator. We denoted by ['XA'nKerQ the [Vafi(lL)]'nKer9. We ‘ ' ", [“JVAT, -- ‘ ‘ y of the h , recall that ['Jf;(U,)]'nKer9 is a subspace of [v.9f;(ll.)]', consisting of the cochains 4) such that 9¢mpmso for arbitrary zo, ..., zv. It is clear that the co-boundary operator 6: [0.)fjl(U,)]'—>[1&fl(U.)]’ carries the subspace [°x;(U,)]'nKer9 into [‘.}€;(U,)]'nl(er9 and accordingly. generates an operator which acts in the corresponding factor-spaces. We write 9!,(p) to denote the kernel of the following mapping:

311(p)= Ker {[°3¢§(U.)]'/[°Jfi.(U.)]' n Ker 9 —’'[13151(U.)]’/[‘-3¢‘§.(U.)II'n Ker 9}.

IV. The Fundamental Theorem

188

We write 24(1)) to denote the family consisting of the spaces 5.07) and their identity mappings. Clearly, 51(1)) is the kernel of the mapping (3: [°X,]'/[°#,]' n Ker 9 —> [‘XlT/[Uf’fl' n Ker 9 .

( 1.5)

We fix an arbitrary integer a. Let (I) be an element of the space 52(11)By the definition of this space, (P is a coset belonging to the factor-space [°.9€(lL)]'/[°;fi(Ua)]’n Ker 9, consisting of the cochains 4):: 422 U, such that 6¢e[‘2fi(U,)]'nKer 9. This inclusion relation implies that the functions

f“(Z)=d‘(z, D) ¢z(Z)|~A,

i=0,

1,

do not depend on Z GOD! and are therefore components ofsome p—function f holomorphic in 9,1. If the cochain (15 itself belongs to the subspace [‘7X;(Ul)]‘ n Ker 9, then f‘50, and accordingly, the function f depends only on the coset «5. Thus, we have constructed an operator d : di—tf, which maps p—functions that are holomorphic in the region {2. into elements of the space 11(1)). We shall call this operator 3 Noetherian operator.

Theorem 1. Let .1! be an arbitrary family of majorants. Then a Noetherian operator defines a family mapping (1: I‘m—mflwfi},

(2.5)

which is an isomorphism. Proof. We fix at arbitrarily. Let (15 be an arbitrary element of the space 5,0,1). We estimate the function f= dd) in the region (2,“. By the definition of the norm in the space 2;,(17), corresponding to arbitrary positive a we can find a representative died) such that 1(¢(l,§ ||d>||1+£. Suppose that ¢=Z¢z Uz. For an arbitrary point Z 69, +1 the function (152 is bounded and holomorphie in the e,—neighborhood of that point. Let ic‘ be the order of the operator d“(z, D) as a polynomial in z and D. From Cauchy‘s theorem, we derive the inequality

mfx |f‘(Z)|§m§1XI I2 ld‘(Z,6)€i|15t¢z(Z)l i g“

§C(|Z|+ I)“ summon. Iz—Zlée.) §C(IZ|+1)" M.+1(Z) ||¢||.§CM.'(Z) ll¢||u Hence

K=max K‘,

oz’=at+l+lc.

||f||./§ C||¢||.§C(II¢II.+E)Since the function f depends only on the coset «if, this equation holds for arbitrary e > 0. and therefore, the term a on the right side can be discarded.

§ 5. The fundamental theorem

189

Thus, we have established that the mapping d:

I,(p)sdi—>d¢ex;.{p,d}

(3.5)

is defined and continuous. The set of all such mappings defines the family mapping (2). We shall show that the mapping (2) is a monomorphism. Suppose that the element @eflflp) lies in the kernel of the mapping (3), that is, d dz = 0 for an arbitrary representative (bed). Corollary 2, §4, implies that 945:0, that is ¢e[°.#;(U,)]'r\Ker 9, whence ¢=0, q.e.d. We construct the mapping inverse to (2). We fix 0:. Let f = {fl} be an arbitrary holomorphio p—function, lying in the space fighkd}. We set U, + 3 = { U2). By definition, U2 is a neighborhood of the point Z of radius a,” $311,. Let the point Z belong to 9,“. Then A Uzc c9“ and therefore, the definition of the holomorphic p—function implies that we can find an elementary covering V of diameter zero and sufficiently small radius and a holomorphic eochain F, of zero order, on the covering Vn4Uz, such that dF=f. Employing CorollaryZ, §5, Chapter III, we find a function it holomorphic on 2 UZ such that 9(6¢—F)=0. We apply to the function d) Corollary 1 of the same section. We have then represented the function :1: in the form of the sum ¢z+pu11, where (it; and W are functions holomorphic in U2, and

supll¢z(1)l, IE Uz} é C(|Z|+1)‘ SUP{r(z)I 9(1) 6) MIN; 152 Uz. mew».

(45)

where s = a, + 3 and r(z) is a power function on some algebraic stratification appearing in inequality (2.3). (The use of the function r(z) in this inequality is legitimate, since we may, of course, suppose that this function does not exceed the function r" appearing in Corollary 1, §4, Chapter II.) In view of the inequality (23) the right side of (4) does not exceed the expression

C(|Z|+1)" mfx SUP lld‘(z, D) ¢(Z)l. ISN‘O4 U2}We remark that d‘(z,D)¢(z)=d‘(z,D)F=f‘(z),

A:0,...,I.

(5.5)

Therefore we arrive at the inequality

sup{|¢z(z)|,zeUz}§C(|Z|+1)"m§1X sup (If‘(z)l. zeN‘n4Uz}~ Since the diameter of the sphere 4U, is less than 9,, we are allowed A to write the inequality

|¢z(2)l


[136$ l Ker 9 —>[1.l’_‘] —»[1x,]‘i[‘x,]' n Ker 9 —>0

.1

.1

.4

0—» ["X‘Tn Ker 9 —> [°.§t’,,]’ ——> Pay/Pm} n Ker 9 —r0 (9.5)

”i

6i

6i

0

0

o

0 —» [15,} (f Ker 9 —. [TAT ——» [my/[my n Ker 92 —» 0

i

§5. The fundamental theorem

191

The exactness of all the rows of this diagram is self-evident. The exactness of the second column follows from Corollary 2, §4, Chapter III. The exactness of the first column is the content of Corollary 3, § 5, Chapter 111. On the basis of Theorem 1, § 2, Chapter II, we conclude that the rightt column is also exact. The exactness of this column in the second term from the bottom implies that the family [JK‘T/[X’AT n Ker 9 is isomorphic to the kernel of (I), that is, to the family 9." (p), and the isomorphism is realized through the mapping (8). Taking account of the isomorphism of Theorem I, we arrive at the isomorphism

[my/[Jm'nKer 9&1”), d}.

(10.5)

Hence it follows that the mapping d in (7) is a homomorphism and an epimorphism and its kernel is the subfamily [affl'n Ker 9. In view of the theorem of § 5, Chapter III, this subfamily is the image of the mapping p in (7), and this mapping is also a homomorphism. Thus, the exactness of the sequence (7) is proved. [I Remark. We shall show that the mappings d and d", which establish the isomorphism (10), have orders independent of the family .1. For this we must return to the beginning of the section. From the proof of Theorem 1, it is evident that the mappings which establish the isomorphism (2), have orders that are independent of .11. We now return to the diagram (9). What we have proved implies that the mappings of the form Ker 6 —. Coim a, which are inverse to the mapping denoted by the symbol 6, have orders not depending on I. For the mapping 0 in the second column, this was established in 3°,§ 4, Chapter III, and for the mapping in the first column, it follows from 9°, § 5, Chapter III. The remark in § 2, Chapter 1, implies that the mapping which establishes the isomorphism

3:401) E [-fitT/[JGJ’ n Ker 9

(11.5)

has a similar property. We remark that the order of 0 also does not depend on .11. The mapping that establishes the isomorphism (10) is the composition of the mappings that establish the isomorphisms (2) and (11). With this the remark formulated above is proved. 3". Supplement to Theorem 2. Our remark allows us to formulate Theorem 2 in a stronger version. We introduce the following notation. The doubly infinite sequence of functions .1! = (MAI), oz= — 1, 0,1,...}, defined in C", will be called a complete family of majorants, if for an arbitrary integer fl, the sequence I,=(M,+,,(z), «=1, 2, ...) is a family of majorants. We shall say that .l is a complete family of majorants

IV. The Fundamental Theorem

192

of type J, if for an arbitrary integer fl, 1, is a family of majorants of type J. The construction of § 1, Chapter III, and also subsections 1° and 2° of this section, can be carried over in an obvious fashion to complete families of majorants 1!. In particular, if .1! is a complete family of majorants, at}, is a doubly infinite family of spaces M2, or: ---, —~ 1, 0, 1, , where 3?; is the space of functions holomorphic in [2,, and having finite norms || - II: (see (1.1), Chapter III). The symbol .916, (p, d} has an analogous meaning. The families affix}, {11,11} and so on, corresponding to the complete family of majorants .11, will be called complete families. The matrix p and the associated Noetherian operator d define the mapping (7), where .1! is an arbitrary, complete family of majorants. We remark that the order of each of these mappings is a function of the form a a at + a, where a is a constant depending only on p and d. In fact, for the mapping p, the constant a is equal to u=deg p, and for the mapping d according to (6), it is equal to 3 + [q]. We shall now formulate the assertion in which we are interested. Supplement. Let .1! be a complete family of majorants of type I. Then the sequence (7) is exact and, moreover, there are defined the mappings of complete families 11“: 9f,{p,d)—>3fj/Kerd, p“: Kerd—nfj/Kerp,

inverse to the mappings d and p in (7), the order of which has the form a —> at + A, where the constant A depends only on p and d. Proof. We fix an arbitrary integer I} and we consider the family of majorants 41,, consisting of the functions .11, M, at: 1, 2, By hypothesis, this family is of type J. Therefore, by Theorem 2, there is defined the mapping 11“: xi,” {1), d) AIL/Ker d, inverse to d. In view of the remark made in connection with Theorem 2, the function ot—>y(oz), which is the order of the mapping :1“, does not depend on the family 11,, that is, depends only on p and d. We denote by 3?; and 1,1“), d} the spaces forming the families .91”,

and an, up, :1). Let

'

(11“)1 2 xi (P. d) -> HGT/(Ker 11);,

(Kerd)§=Ker{[9fi’]'—‘> Advil},

v=v(1)

beIthe component of the mapping d“. We remark that the spaces at”, {12,11} and 3?; are also components of the families 1:, (p, d} and

§5r The fundamental theorem

193

x‘ respectively, With the indices 5 + l and [3 +7. Therefore, the mapping

((1‘1)’l can be rewritten as: 3%+1lptd}->x§'+y/(Kerdh+y~

We denote this mapping by M“), H. Since the composition of the map pings (Ll—1); and d is the identity mapping, acting from m’md} to 1/;n d}, and since the mapping d, defined on [MT/(Ker 11);, is biunique, the mappings (11“), form a mapping of families d“. This is the one we are looking for, since its order is equal to the function at —+ at + y— 1, where y=y(l) is a quantity depending only on p and d. In a similar fashion, we construct the desired mapping p‘l. [1

4°. The invariant notion of holomorphic p—function. By Definition 1, the concept of holomorphic p-function depends not only on the matrix p itself, but also on the choice of the associated set of normal Noetherian operators. We shall now show that in fact the content of this concept depends only on the 3Lmodule M gCoker p=9‘/p 97". We remark that in View of Theorem 1, the family #1,”, d} of the spaces of holomorphic p-functions is isomorphic to the family 54(1)), which does not depend on the construction of Noetherian operators. We shall show that for an arbitrary finite 9-module M and for an arbitrary family of majorants .II, the families 331(p) with the matrices p such that Coke: n are connected among themselves by natural isomorphisms. Proposition 1 I. Let .1! be afamily of majorants and let f: M—>l\/l’ be a mapping of finite W—modules with M $971)?“ and M’zy‘yp’?‘ for some 9-motrices p and p’. Then there is defined a mapping offamilies f; : Q'Ap)» 5.410),), depending only on L p and p’, and satisfying thefollowing condition. 11. Iff is an isomorphism and p=p’, then f;' is also an isomorphism.

111. If

F M" 1 M A» M’

is a commutative diagram of mappings of finite 9-modules, then for arbitrary 9-matrices p, p’, p", the cokernels of which coincide respectively with M, M'. and M”, the diagram

”5,. elm") 4;.

is also commutative.

moi—’F—enp')

194

IV. The Fundamental Theorem

Proof. Suppose given the mappingf: M —> M’. We construct the mapping fp". In view of Proposition 1, {53, Chapter I, there exist 9-matrices f0 and 1”,, such that the diagram

flaw—tw—m

I.)

1..)

J

(12.5)

9- —P» 9' —, M —»o is commutative. But its commutativity implies the relation f0 1;: p‘ 1”,. Hence, it is easy to see that the operation of multiplication by the matrix fo in the space 0‘, (the point ZeC’I is arbitrary) carries the subspace p 0‘, into the subspace p’ 0‘,’ :02. For an arbitrary integer a the operation of multiplication by the matrix f0 defines a continuous mapping

[’JflUJJ’ —' ['Jf‘émwumflra

(13-5)

where m is the order of this matrix. Let 9 and 9’ be p- and p’-operators respectively. If the cochain 4) belongs to the subspace ['1;(U,)]’ n Ker 9, then by the property of the operator 9, for arbitrary ZEC’l the cochain ()3 belongs to p 0‘; (that is, each of its components belongs to p (9}). Therefore, the cochain f0 ()5 belongs to [VM+M(U,+M)]" n Ker 9’, Thus we have established the fact that the mapping (1 3) carries the subspace [bat/QT n Kerg into ['Jfimwnmfl" n Kerfl’ and, therefore, defines a mapping of the corresponding factor-spaces. The set of all such mappings forms the family mapping

[0: V 4/” ,nKu9~Vfl/inxa9'.

(14.5)

Clearly, mappings of this kind commute with the coboundary mappings of the form (1). Therefore, the mapping (14) carries 54,01) into 51(1)), that is, it defines the desired mapping

fp'": 5AM—r-‘2'AP'L We shall show that this mapping depends only on f, p and p’ and does not depend on the matrices f0 and fl. An arbitrary pair of matrices (f0, [1), which render the diagram (12) commutative, will be said to correspond to the mapping f. Let (fo,f1) and (g,,, g) be two pairs of matrices corresponding to the mapping f. In accordance with Proposition 1, § 3, Chapter I, we have the equation f°=go + p’ n, where u is some 9-matrix of size s’x t. Since the image of the mapping [7’ n: V j, —> W; belongs to the subfamily p’ W}, in going to a mapping of the type (14)

we obtain a null mapping. Hence, 13:12,, q_e.d,

§ 5. The fundamental theorem

195

We shall establish Property II. Let the mapping f be an isomorphism and let p = 11’. Then we may choose for fa and f, the unit matrices. Clearly the associated mapping f; is an identity mapping. We shall establish the third property. Let (/0, fl) and (g0, g) be pairs corresponding to the mappings f and g. From the commutativity of the diagram 9’"L">9""—>M"—v0

4 wt 4

t?" it?" —>M' —>0

,4 h} 4

9'4»? —vM —»0 it is obvious that the pair (gofl,,g1fl) corresponds to the mapping gf= h. By definition, the mapping hg" is associated with the matrix go f0 and, accordingly, is the composition of the mapping f: and ggi', associated with the matrices f0 and go. [I

Suppose that the mapping f: M —>M' is an isomorphism. Then our proposition implies the existence of the mapping

so (”—74% 33.020,

(15.5)

which establishes the isomorphism of the families .9!1(1)) and 210’). Thus for arbitrary pairs of 9-matrices p and 12’ such that Coker pg Coker p’ a M, we have established an isomorphism of the families 3,, (p), and 3,, (11’). These isomorphisms agree in the sense that for an arbitrary three matrices p, p’ and p", the cokernels of which are isomorphic to M, the diagram

F I; (11") r—l

2'40?) r—* 5401') which is formed by these isomorphisms is commutative. (Its commutativity follows from Property III.) For an arbitrary finite fl-module M, we regard all the families 3:401) for which Coker n, and isomorphisms of type (15), as a new object, which we will denote by the symbol 1AM). Isomorphisms of the type (15) will be referred to as inner isomorphisms. Let f: M —» M’ be a mapping of finite 9-modules. By Proposition 1, for any matrices p, q and p’, (1’, whose ookernels are isomorphic to M

196

IV. The Fundamentnl Theorem

w

and M', the diagram

1'; (P) —f’-' 5224 (11’)

I

I

alto—{Leena in which the columns contain inner isomorphism is commutative. The set of mappings ff will be regarded as mappings of the corresponding objects. We now take the final step. For every finite 9-modu1e M, we shall consider the new object at”, {M}, formed by the family 101(1), d}, where Coker p; M. These families are interconnected by isomorphisms, which are compositions of isomorphisms of the type (2) and (15). If we are given a mapping of modules f: M aM’, selected matrices p and p’, the cokemels of which are isomorphic to M and M’, and if we have constructed the

corresponding Noetherian operators 1! and d’, then we can construct the mapping [1575“: ”11(1), 11} —> .13“); d’}, which is defined by the condition that the diagram

,, ,. ”14 (P, d} #1 ”xii, 1'}

”I

rm»

"l

—""—»z..M’—’>M—‘>M”—>0.

ivQfififio 1‘. in

o—>§—e§—»§i—ro

is exact. By the lemma of § 3, Chapter I, such a sequence can be imbedded in a commutative diagram of the form 0

l l . _. M —»0 i l

"'—>M"—>0

—>M'—>0

0

(19.5)

N The Fundamental Theorem

198

in which all the rows and columns are exact. We consider another diagram

0

0

0

i i i i i i 1;, A31; 4.1mm;

xg'Asa'flxflip".d"}—»o

i

I ,

—.0

(20.5)

i

x; —"’»s€’;.’—‘>amp'.d'} —»o

i

i

0

0

0

From the exactness of the columns of diagram (19) and from the theorem of §5, Chapter III, there follows the exactness of the columns of diagram (20), except the rightmost. The exactness of the rows follows from Theorem 2 of this section. Applying Theorem 1, §2, Chapter I, we establish the exactness of the right column. In View of the remark made at the end of 4°, we may conclude that the sequence

0—» X,{M'} A MAM} in {M”} —.0.

(21.5)

is exact. Now suppose we have an arbitrary exact sequence (17). In view of the exactness of the sequences

OaCoimf—RM —I>Img—*0, 0—»Kerf —>M">Coimf—>0,

0—»Im g—>M"—»Cokerg—>0 the sequences 0—»Jfl{Coimf}##4{M}—>X’A{Img}—>0,

J?" {M’} —v #1 {Coimf} —v 0a #1, (1m g} —> 1",, {M”}.

are also exact. Hence, it follows that the mappings

fa: JVAM’} exnlCoimfP'faiM). 24.1 fllfixaiIm gl—MYAM") are homomorphisms and Im fI~Ker g4, (that is, these properties are enjoyed by the corresponding representatives of the sets I, and g1,). l]

§ 5. The fundamental theorem

199

6°. Remarks Remark 1. Theorem 2 is a particular case of the Fundamental Theorem, corresponding to the exact sequence ?'—’>.9’—>M—»0. Remark 2. We shall formulate a sharpening of the Fundamental Theorem. Let f: M —> M' be a mapping of finite modules, and let p and p’ be 9-matrices, whose cokernels are isomorphic to M and M’. Then for an arbitrary family of maj orants .ll oftype J, the mapping/If)" : it}, (p, d} —> if,“ {p’, d') is a homomorphism and the order of the mapping and its inverse does not depend on .11. The proof of this assertion is left to the reader.

PART TWO

Differential Equations with Constant Coefficients Chapter V

Linear Spaces and Distributions In §1 we continue the study of families of linear topological spaces that was begun in § 1, Chapter I. We consider two special types of spaces: Frechet spaces and Schwartz spaces. Our attention is mainly focussed on inductive and projective limiting processes. In § 2 and 3 we set forth the theory of distributions and Fourier transforms in a form suited to our needs. § 1. Limiting processes in families of linear spaces By a linear space, or more simply, a space, we shall mean a linear topological locally convex space — we shall use the abbreviation 1.t.s. — over the field C of complex numbers. We remark that this concept is a special case of the notion of a topological module in the sense of §1 Chapter 1‘ We now introduce families of l.t.s. of a more general type than those in §1 Chapter I.

1°. Increasing families of I.t.s. The set .521 is said to be a directed set if there is an order relation among its elements, X, , defined for arbitrary a’ and agat’, and satisfying conditions like (i) and (ii) as given in Definition L Every decreasing family of l.t.s. has a projective limit, defined as follows. A thread is an arbitrary set of the form x={x,,eX,,ae.n!}. in which the elements xcl satisfy the relations if x,=x.» for arbitrary at’ and aga'. The set of all threads will be denoted by X. Threads may be

§ 1. Limiting processes in families of linear spaces

203

multiplied termwise by scalars, and added termwise; the set X is therefore a linear space. We denote by 1“: X —v X the linear operation which maps the thread {x,) into the element x. This will be called a canonical mapping. We now introduce a topology mto X, the weakest of all topologies for which the mappings1 are all continuous. A neighborhood of zero in X is defined as an arbitrary set containing a set of the form (1“)“(U,), where U, is a neighborhood of zero in X,. The l.t.s. so constructed will be written in the form

X=1im lim {x,, if}. 4— X= X, which IS obviously continuous Let us construct its inverse. Let x,eX be an arbitrary element of UX,. We choose an

1 x belongs to U X and ls equivor’ed’ such that ot X; both are obviously continuous [I

V. Linear Spaces and Distributions

204

4°. Countable families. We shall often encounter increasing and decreasing families of l.t.s. defined on the set Z + of natural numbers, ordered by increase in value. Another particular case that will be of importance to us is the family in the sense of § 1, Chapter I, an increasing family defined on the set Z of all integers. The set Z has two order relations — by increasing value and by decreasing value. In both cases, Z is a directed set. Thus, every family of l.t.s. in the sense of §l Chapterl is both an increasing and a decreasing family, and therefore has both an inductive and a projective limit. We remark that every increasing or decreasing family of l.t.s. defined on the set Z + can be looked on as a family in the sense of § 1 Chapter I. For, if {X,, If} is an increasing family of Lt.s. defined on Z+, we may supplement it with null spaces X“ and null mappings i": for all integer a§0; we obtain an increasing family defined on Z. If {X,, if} is a decreasing family defined on Z +, we may set

Y

X_,,

,— 0,

aE" defined by the formula (¢‘f’, e)= (f’, 4; e) is also continuous. Let X ={X,,i:'} be an increasing family of l.t.s. We consider the ensemble of conjugate spaces X: and conjugate mappings

fi'=(i:,)*: X:—>X,‘5. These spaces and mappings clearly satisfy the conditions given in Definition 2 and therefore form a decreasing family, which we call the conjugate family and denote by X*. Similarly, for every decreasing family X we define the conjugate increasing family X*. Suppose that X ={X,, if} is an increasing family, and that B, is a bounded subset of one of the spaces X,. Its image i485): X is obviously absorbed by an arbitrary neighborhood of zero in X and is therefore bounded in X. We shall say that the family X is regular if the converse assertion is true: every set bounded in X is for some at equal to i,(B,), where B, is a bounded subset of X,.

Proposition 3. Let X be a regular increasing family. Then there exists a natural isomorphism

(X)*‘=“X‘=li_mixf.i:'=(l‘2-)*}~

(3-1)

Proof Let f be an arbitrary continuous functional on X. Since the canonical mapping i,: X,—> X is continuous, we may consider the continuous functional f,=i:fEX}. The ensemble of these functionals is obviously a thread in the family X* and therefore is an element of the limit X*. Conversely, let {LEXI} be a thread which is an element of X'1 We consider the functional defined on X by the formula

(fl i.X.)=(L.Xi).

MEX..-

It is clearly linear and continuous, and is therefore an element of(X)". But this establishes the algebraic isomorphism (3). We show that this isomorphism is also topological. By definition, the polars of the bounded‘ sets BCX form a fundamental system of neighborhoods of zero in (X)*. On the other hand, in each of the spaces X: a fundamental system of neighborhoods of zero is formed by the polars of the bounded sets Bac. Therefore, the sets of the form (f‘)"(82) form a fundamental system of neighborhoods of zero in X“ (j’ is the canonical mapping in the family X‘). We now note that (j‘)‘ l(Bl?) = (i, (3,3)". There remains to be added the fact that by hypothesis the classes of sets of the forms B and MB.) coincide. [J Proposition 4. Let X be an‘increasing family consisting of reflexive Panic: spaces. Then the limit X is also reflexive and the isomorphism (3) is va 1 .

§ 1. Limiting processes in families of linear space:

207

If X is a decreasing family consisting of reflexive Banach spaces, the limit X is also reflexive and there exists an isomorphism (X)“ E X*. Proof. As we know, an increasing family consisting of reflexive Banach spaces is regular and its limit is reflexive 1. The isomorphism (3) follows from Proposition (3). Suppose that X is a decreasing family of reflexive Banach spaces. The conjugate family X* is an increasing family and also consists of reflexive Banach spaces. By what we have already shown, the limit X* is reflexive and (X‘)“=X. Taking the conjugates of both sides of this isomorphism, we obtain the isomorphism (XVk = X‘, q.e.d. 1] 7°. Schwartz spaces. We recall that the mapping 45: E —>F of l.t.s. is said to be compact, or completely continuous, if it carries a neighborhood

of zero into a relatively compact set 3. If E and F are Hilbert spaces, the conjugate mapping 4:": F*—>E* is compact if and only if 45 itself is compact. The l.t.s. E is a Schwartz space if, corresponding to an arbitrary neighborhood of zero U in E, we can find a neighborhood V of zero which for arbitrary e>0 admits a finite c U-net, that is, can be covered by a finite number of translations of the set a U. Our main concern is with Schwartz spaces that are also Frechet spaces, which we refer to as Fly-spaces; we shall encounter them as projective limits. Proposition 5. Let X ={X,,i,','} be a decreasing family having the following properties: for arbitrary or, X, is an f-space, the mapping 1:“ is compact, and its image is dense in X,. Then X is an fig-space. Proof. By Proposition 2, X is an f-space and for arbitrary or the image of the mapping I": X —> X, is dense in X,. Let U be an arbitrary neighborhood of zero in X. It contains a neighborhood of the form (i')“(U¢), where U. is a neighborhood of zero in X,. Let U,“ be a neighborhood of zero in X”, such that the set iLdU. +1) is relatively compact in X“. We set V=(i‘+‘)“(UM 1) and we show that the neighborhood V, for arbitrary e> 0, admits a finite a U-net. Since the set fi+1(U,+1) is relatively compact in X," it belongs to a finite union of the form A s

U (x, +7 0.), A

where all the xfieX,. We choose the elements x15)? so that for all 1. we a have i“l x“ — x: e 7 (1.. . . ~ 2 See Kantorovic and Akilov [l]. 3 The set K in the l.t.s. E is said to be relatively compact it Its closure is compact.

208

The“

V. Linear Spaces and Distributions

13+;(IL+1)CU(I“X‘+8UJ~ 1

Applying to both sides the operation (1")'1 we obtain the inclusion VcU(x‘+e U). But this constructs the finite e U-net for the set V. [] We note that every fy-space is reflexive. For, by a general criterion ‘, for an f—space X to be reflexive it is suflicient (and necessary) that every closed bounded set B be weakly compact. If X is an Sty-space, then for every neighborhood of zero U there exists a neighborhood of zero V which admits a finite a U-net for any e>0. Since B is bounded it is contained in J. V for some A> 0 and consequently has a finite U-net. The space being complete and B being closed, B is compact and accordingly weakly compact q.e. d. Let E be an 1‘ t. s. and let F be a subspace of it. The polar of the subspace F is a subspace F0c 5* consisting of those functionals that vanish on F. If E is reflexive, the second polar F0° is a subspace in E, and F CF°°. We will have F =F°° if and only if F is closed. Let F be a closed subspace of E. We shall establish the following algebraic isomorphisms 5: F*;E‘/F°,

F°§(E/F)‘.

(4.1)

By the Hahn-Banach theorem, every continuous functional on the subspace F can be extended to a continuous functional on the whole space E. The extension is defined up to a functional vanishing on F. But this defines a linear mapping F" -> E“/F°. Conversely, every element of the factor-space E"'/F° can be looked on as a continuous functional on F. This establishes the first of the isomorphisms (4). Let us construct the second. To every functional fEF° we may relate a functional f on the factor-space E/F, which we shall call the functional associated to f. Conversely, every functional on E/F may be looked on as an element of E‘ that vanishes on F. The significance of the iY-spaoes is brought out in the following proposttion. Proposition 6 6. Let E be an fY-space, and let F be a closed subspace of it. Then F and E/F are fly-spaces, and the algebraic isomorphisms (4) are topological isomorphisms. . We note that under the hypotheses of this proposition all the spaces in (4) are reflexrve. Therefore if we pass to the conjugate isomorphism s 4 See, for example, Bombaki [1]. Chi IV. §3, No.3. 5 We recall that unless we explicitly state the contrary case, we endow every subspace F :5 With the induced topology. and every factor space E/F with the canonical topo ogy. 6 For the proof see Grothendieek [l].

§ 1. Limiting promos in families of linear spaces

209

we obtain F :(E"/F°)*, (F°)*§E/F. These isomorphisms show that Proposttioné holds also when E“ is the conjugate of a Schwartzian fY-space and F0 is an arbitrary closed subspace of it. 8". Duality in exact sequences of Lt.s. We recall some definitions from § 1, Chapter I. A continuous mapping ()3: E —> F of Lt. s. is said to be a homomorphism if the associated mapping ¢ : Coim d) = E/Ker 4) —> Im ¢ is a topological isomorphism. The mapping ()3 will be a homomorphism if and only if the image of an arbitrary neighborhood of zero U CE is a neighborhood of zero in Im if). The sequence of mappings of 1. t. s.

E A» F 4*. G

(5.1)

is algebraically exact if lm¢= Ker 11/. This sequence is said to be exact if it is algebraically exact and the mappings 43 and 1/; are homomorphisms. The mapping up in this sequence is called an (algebraic) monomorphism if the sequence (5) is (algebraically) exact and 41:0. We call 45 an (algebraic) epimorphism if (5) is (algebraically) exact and 11/ = 0. In other words, to say that a mapping ()5: E a F is an algebraic monomorphism means that Ker ¢=0, and to say that it is an algebraic epimorphism means that Coker ¢=0. A mapping 43 is a monomorphism (epimorphism) if and only if it is an algebraic monomorphism (algebraic epimorphism) and also a homomorphism. Proposition 7 I. Let E and F be f—spaces, and let ()5: E —>F be a continuous mapping. Then the three following assertions are equivalent: a) ()3 is a homomorphism; b) the subspace Im (15 is closed in F; b“) the subspace Im «15* is closed in E“; II. If E and F are FRY—spaces, these assertions are also equivalent to the following: a“) (15* is a homomorphism. Proof. The equivalence of the first three statements is knowri7. Let us prove the assertion I]. We suppose that the conditions a), b), and b“) are satisfied. Condition a) means that the mapping (5: E/Ker 45 —> [In (it associated with d) is an isomorphism. Therefore the conjugate mapping (W: (Im ¢)‘ —s (E/Ker ¢)* is an isomorphism. Proposrtion 6 implies the two isomorphisms (Im ¢)“;F"/(Im ¢)° (since Im q?“ 15 closed) and (E/Ker ¢)‘;(Ker ¢)°. Taking account of these two isomorphisms and

the equations (Im ¢)°= Ker ti" and (Ker ¢)° = Im (13" (smoe Im ¢* 15 7 See Dieudonne and Schwartz [1}

Vt Linear Spaces and Distributions

210

closed), we may rewrite the mapping (JV in the form F*/Ker¢* —> In: 41*. We easily see that this mapping is associated with (V; this establishes . that d)* is a homomorphism. Conversely, suppose that (15* is a homomorphism, Le. that the associated mapping 5’: F‘/Ker ¢*—>Im (11" is an isomorphism. By Proposition 6 the space conjugate to F*/Ker da” is isomorphic to a closed subspace (Ker ¢*)°cF and therefore it is an fV-space. Since it is reflexive, its conjugate is F'/Ker 43‘, which is therefore conjugate to an f—space and consequently complete. Since Im 45" is isomorphic to F‘/Ker ¢*, it is also complete and is therefore a closed subspace in E‘. This proves b‘). I]

Proposition 8 I. Suppose that

ELF; G

(6.1)

is a sequence of continuous mappings of Lt.s. If it is algebraically exact, and if [b is a homomorphism, the conjugate sequence

5* «—“ r* Y is an ensemble of continuous linear operators «5,: X,—’ Y”), defined for all integer a, and such that for arbitrary a and a’>a the commutative relations 115,. i;'= fig} dz. are satisfied. The function mafia), which we call the order of the mapping ()5, must be monotonely increasing and must tend to -t_- 00 With on. If 4): X a Y and 1/1: Y—>Z are two family mappings with the respective orders [3(a) and y(oz), their compositions is the mapping W 4): X —»Z with the order 70301)) and the components 1/1,", (that, and 42, are the components of III and d3). The identity mapping I: X —>X is a mapping with components of the form 1:"

Proposition 9 I. Let 4): X —> Y‘be a famil mapping. Then there exist uniquely defined mappings (£3 X —> Y and : X —> Y such that for arbitrary a the following diagrams are commutative:

X i» )7 :4

If:

X, A, y,

X i» y

1-1

1- B=fl(a),

(8.1)

X“ A Y,

where i,, j,, i" and j" are canonical mappings. The mappings a; and (5 are called the limits of a). II. If 41, ¢’: X —’ Y are equivalent mappings (see §1 Chapter I) their limits coincide. III. For‘any two mappin s 4): X—rYand I11: Y—>Z we have the relations Wait} and ”:17; Proof Let us establish the first assertion. The function I]: a—vma), ie. the order of d), is by hypothesis monotone and tends to ice as a—vioo. Therefore the image of the mapping [3: Z —»Z is cofinal in Z for both the ordering relations in Z discussed in 4°r Let us consider the corresponding eofinal subfamily Y’={Y,,’= YWO). The mapping ¢ can be considered a mapping from X to Y', and its order is the identity function aaa. 0n the other hand, by Proposition I, both the limits of the subfamily Y’ are isomorphic to the limits of the family Y Therefore we may suppose from the outset that [3020501. Let us construct the mapping $. The diagram (8) will be commutative if and only if it carries an arbitrary element of the form i, x. into Lo, x,. We shall show that this condition is correctly formulated, i.e. show that the element j, o, x, does not depend on x., but only on i, x,. If

212

V. Linear Spaces and Distributions

i, x,=i.. 11,1 there exists by definition an Macao! such that tf'x,=1§7x,, whence

i: ¢.X.= 15,11: X.=¢1v1371.x =f' 1131):“ that is, the elements ¢,x, and 45,l1 are equivalent, and therefore j, d). xfl= j, 41,, x“ q. e. d. We have thus proved that there exists a unique mapping which makes the diagram (8) commutative. The continuity of (i follows from the definitions. Now let us construct the mapping ¢. Let x: {x,} be an arbitrary thread belonging to X. The elements 45,): EY also form a thread. This follows from the computation

fi'¢axa=¢.rlfi'xa=¢alxau

a’>a-

We write $x={¢,x,,}. This definition of (13 is necessary and sufficient for the commutativity of the diagram (8). It is easy to prove that the mapping is continuous. Then the second assertion is proved. The third assertion is obvious. It is also obvious that the limits of an arbitrary identity mapping are identity operators. Now suppose that 111,113: XHY are equivalent mappings. By definition, this means that there exist identity operators J, J: Y—bY such that J 45: J’ ’.It follows from what we have said that .17: .143: $, whence d3: .Similarly, I] We note some consequences of Proposition 9. Let I: X —>X be a unit mapping, i. e. a mapping whose components are the operators 1:.' We apply our proposition to it. Since (8) is commutative, the limit mappings I and I are identity mappings. Now let I be an identity mapping of the family X. It IS equivalent to a unit mapping and therefore Assertion 11 of our proposition implies that the limits I and I are also identity operators. Let us suppose that the mappings ¢z X —» Y and III: Y—> X establish an isomorphism of these families. We shall show that the limits

1&9,

1&1

are also isomorphisms. By hypothesis, 1/1 41: I and 4: 1/1: J where I and J are identity mappings Using Assertion III, we find that W —1[I¢ and (W: (511/. On the other hand, it follows from what we have said above that the mappings W=I and “=1 are identity operators. Hence it follows that the operators 4) and 1/: are mutually inverse. Similar

arguments will show that the operators 115 and J: are also mutually

inverse. Proposition9 proves that the operation of passing to either the inductive or projective limit is a functor acting from the category of

§ 1. Limiting processes in families of linear spaces

213

classes of equivalent families of 1.t.s. and the classes of equivalent mappings of these families to the category of linear topological spaces. We now prove two propositions characterizing the exactness of these functors. 10°. Passage to the inductive limit in exact sequences Proposition 10. Let

0 —+ X A» Y—‘» z —> 0, . . , X={X.. 17.}, Y=(Y.,JZ ). Z={Z..kZl

(9-1)

be a sequence of increasing families and let

Daft—‘n‘l—flz—to

(10.1)

be the sequence of their inductive limits. A) If the sequence (9) is algebraically exact in the term Y, the sequence

(10) is algebraically exact in the term Y. B) If the‘sequence (9) is exact in the term Z, the sequence (10) is exact in the term 2. Remark. Assertion A) implies that a passage to the inductive limit conserves algebraical exactness in an arbitrary sequence of increasing families. We precede the proof of the proposition by a few general remarks. We shall suppose that (9) is an algebraically exact sequence of families. I. The exactness of (9) implies that ‘1’ ¢ ~0, i.e. there exists an identity mapping K of the family Z such that K a ¢=0. Replacing the mapping up by the equivalent mapping l//’=K¢ we obtain the relation ll/’¢=0 and still maintain the exactness of (9) and do not change the limit mappings ;5 and IL. Therefore we may suppose that the original mappings satisfy W (1) =0. II. Suppose that 4": X‘s—’Ymu)»

'f’afi X1421“)

are the components of the mappings 45 and ill. By hypothesis [3(a) and you), the orders of these mappings, are monotone increasing and tend

to :t 00 with at. The composite function y(fl(a)) has the same property. It follows that the subfamilies

1" = { Kohl-5w} v

2': (limo), 14:512.?)

are cofinal respectively in the families Y and Z. If we replace Y and Z in (9) by Y’ and Z', the orders of the mappings it and w become the

214

V. Linear Spaces and Distributions

identity mapping 0: —> a. We may therefore suppose from the outset that 5 (or) E y(a:)E 0:. 111. By definition, the exactness of(9) implies that there exist mappings 1/)“: Kerw—tX,

1/1": Z—>Y/Ker1/I

(11.1)

and identity mappings I, J, K of the respective families X, Y, Z such that

Ker¢cKerI,

¢¢"=J,

1/11/I"=K.

(12.1)

Let a(a1), box), 2(a) be the orders of the mappings I, J, K. We note that the orders of 1/)" and 1/1“ are b(11) and C(01) respectively. We choose some monotone increasing function 1(a), tending to i 00 with a, which for arbitrary u satisfies the inequalities

a(A(rz)), bung), c(l(a))§/l.(at+ 1).

(13.1)

If we replace the mappings I, J, K, 45“, 111—1 by their compositions with suitable identity mappings, we may so increase their orders 2(a), box), and C(01) that the inequalities (13) become equations. Relation (12) is not disturbed Next we replace X Y, and Z by the cofinal families {X1110}: {Yni’}, {21”,}. The result 15 that the orders of all the mappings I, J, K, 115“ ,‘11// are now equal to the function u»a+1. We may therefore suppose that the mappings I, J, K, 43", d1 ‘constructed for the initial sequence (9) have the same property. ProofofProposman 10. We shall establish Assertion A). The equation 1/143: 0 implies (see Remark 1) that 1/15: 0, whence ImcficKerlfi. Let us prove the converse inclusion. Let y be an arbitrary element of the space Ken/t. Since ‘

Y= U MY), where j, is a canonical mapping, we have y—1,, y,‘ for some yleY. The equation 1/ y: 0 and the relation 1/4,: k, 1/, (see Remark II) imply that k 1/1,,y,=0. This means that k‘u/I,y.= —.0 for some fiZa. Since k‘1/1,y,, 1;,j2y, the element jfly, belongs to the kernel of 1b,. The relations (11) and (12) imply that the identity mapping 1 carries Ken/I into Im¢, and therefore its component jg“ carries Ker 1/], into Im 43,“. Hence jfl” y,=—11$‘,+,x‘,+l for some Xp+1EXp+1- Therefore y= (fix, where x= ln+1 x5“, ‘1- e. d~ Let us now rove Assertion B). By hypothesis the mapping associated to 1/1, namely : Y/Kerv/1—>Z, is an isomorphism. The limit mapping

.13: msz

§ 1. Limiting processes in families of linear spaces

215

is an isomorphism. by Proposition 9. Let 7t: Y—> Y/Ker (1/ be a canonical mapping of the family on its factor-family, and let it: Y—Hm

be its limit. Since Ib=lll 1r we have ub=lb it. Thus, we have only to show that 7‘: is a homomorphism. This means that the image of an arbitrary convex neighborhood of zero V in l7 is a neighborhood of zero in Y/Ker b. Suppose that for every :1 the mapping 1:“: Kan/Ker fl is a component of the mapping 1t, and that 1”,: Y,/Ker wflaY/Ker is the canonical mapping. The equation

fifl.i;‘(V)=ijui;‘(V)=i(V) implies the relation

1r.(iI‘(V))C(IZ)“(1r(V))~

(14-1)

Since by hypothesis we have for every a that j; l(V) is a neighborhood of zero in Y,” and the mapping an is a homomorphism, the left side of (14) is a neighborhood of zero in Y./Ker up... Thus, for arbitrary at the set (11)” (1t(V)) is a neighborhood of zero in Y,/Ker lbw Therefore the set 1r(V), being convex, is a neighborhood of zero in Y/Ker E5. I]

11". Passage to the projective limit in exact sequences of families Proposition 11. Let (9) be a sequence of decreasing families and mappings, and let

oak—tiiLZ—m

(15.1)

be the sequence of their projective limits. I. If the sequence (9) is algebraically exact in the terms X and Y, and (17) is algebraiplu'sm, then the . ,r' a 4) is a ’ if the cally exact in the terms X and 7, and the mapping if is a homomorphism. 11. Suppose that the sequence (9) is exact and that it satisfies the following conditions: a) for arbitrary or, Xrx is an f-space; b) for arbitrary or the image of the mapping 1‘; +1: X, +, —> X, is dense in X,; Then the sequence (15) is also exact. Proof. Since a decreasing family is also a family, we may adopt the stipulations I, II, and III in 10°, replacing 01+] by a—l in III. We must note that these stipulations do not violate the conditions a) and b), since to adopt them we must replace the families in (9) by colinal families.

216

V. Linear Spaces and Distributions

Such a substitution obviously does not violate condition a). Since an arbitrary mapping 12' is a composition of the mappings i5", the condition b) implies that the image of any if is dense in X,“ Therefore the condition b) is also conserved under a passage to a cofinal family. Let us prove the first assertion of our proposition. We first show that {3 is biunique. Let x be any element of the space Ker (i, and let {x,} be the corresponding thread. By hypothesis ¢,x,=0 for all at. Because (9) is algebraically exact in the term X, we conclude from the first relation in (12) that i:“x,=0 for all at. Since (x,} is a thread, we have tZ“x,=x,,1, and therefore the thread consists of zeroes, i. e. x=0, q.e.d. But this proves that (15) is exact in the term X. Let us now show that (iv is a homomorphism Since 4: is a homomorphism, and since (9) is algebraically exact in X, the associated mapping 6: X jlmdz ‘is_a_family isomorphism. By Proposition 9 the limit mapping (5: X —>Im¢ is an_isomorphism of spaces. But :fiiis the composition of the isomorphism (5 and the natural mapping 2: 1m (1) —> Im 15 Thus there remains only to show that e is also an isomorphism. The spaces Im 43 and Im t} coincide algebraically and e is an identity mapping. Therefore we need only prove that the tMog-ies on these spaces coincide. Let 1) be a neighborhood of zero in Im 42. By definition it contains a set of the form (1')"(v,), where v, is a neighborhood of zero in Im 42,, i.e. a set of the form V, n Im (1),, where nweighborhood of zero in Y,. If we think of j' as a mapping from Imd) to Im (15, we obtain the relations

v:(1")"(v.)=(i')_‘(Kfilm¢.)=(i')‘l(V.)The set (f‘)‘1 (V,) is a neighborhood of zero in Im (5. Conversely, every neighborhood of zero in Im (5 contains a set of this form, and therefore corfls a set of the form (1‘)“(v,), whicfla neighborhood of zero in Im 45. This proves that the topolgies in Im (I) and Im a} coincide, and this, in turn, completes the proof that (5 is a homomorphism. To complete the proof of the first assertion there remains to be proved the equation Im $=Ker h. The equation w¢=0 implies that $=0, i.e. Im (ficKer J1. We must prove the converse inclusion. Let y: {y,} be an arbitrary element of Ker III. From (1 l) and from the second relation in (12) it follows that the mapping J, which is of order a —w a- 1, carries the subfamily Kemp into [m 42. Therefore, for arbitrary u the element y, =1: + , y, +1 belongs to [m dz,“ that is, it is equal to dz, x,, where X,EX,- We shall show that the elements x;=i: +1 x,H form a thread. For any a thug—iii“ xa+i)=¢a x,—j:+‘ ¢a+i xn+l=ya_j:+l Yx+t=0,

§l. Limiting processes in families of linear spams

217

since the elements y, form a thread. Thus the dill'erence x —1,,+ 1 x“+1 belongs to the kernel of ¢ and so to the kernel of 1‘; ‘ in accordance with the first of the inclusions 1n (12). Hence X;_1—l: 1x;=ln_(xu—i:+l xz+l)=07

i.e. the elements x; form a thread, which we shall denote by x’. The relations , 1» 1 _1 ¢u—1 Xa—1=}a

¢nxu=jz

Yu=yu—1

imply that (fix’z y, i. e. the element y belongs to 1m é, q. e.d. This com— pletes the proof of the first assertion. Now we pass to the second assertion We suppose that the metric flu in X, is a nondecreasing function of a: (of. Proposition 2). We fix e>0 andl an element 2: {z }eZ. For any at we choose yue ‘1‘, 2,“, where 11/;11 is the component of the mapping W‘l By the third relation (12) we have 1/: yfl=z . Hence

WAY —j:+1.V.+1)= ‘1’]. ya_ 1‘2“ ¢+lyu+l=za_k:+lza+l=0 Le. yu—j; +1 Y. +1eKer 1/1.. By the second relation (12) ifl maps Ker up, in 1m ¢._,. Therefore for any a>l we obtain the inclusion

1‘1" 14- y;_1elm ¢._1

(16.1)

where y;=1:+1 y.+1~ We shall construct a sequence of elements yte satisfying for arbitrary [1’ the conditions

(i) (ii)

yf-ylelm '15, xn—IEXn—b

j5_1Y;—Y7—1=¢p—1(xpet)v e

flp—1ix1_1y0)§7-

with yf=y;. Let us suppose that we have constructed the elements y; with [f g at and we construct the element y; 1. The inclusions (16) and (i) imply that {uGXm j:+1}’ix+1—Y:=¢u(€.),

By condition b) there exists an element 5,, +1EX“! H such that 5

Pa (x_) é W

where X.=€.-l‘:+1§.+1. We 56' y:+1=J/.+1—¢..+1(§.+1) and verify (ii): )2“ Y:+1—Y:=}:+1 Yin-Y: ‘fiu ¢n+1(§a+t)=¢a(xa)-

So, the sequence {yt} has been found,

218

V. Linear Spaces and Distributions

In view of (ii) the series xa+iz+lxn+l+"' +l7xr+m converges in X, for every or since X, is complete. Let x‘ be the sum of this series. Since X, is a separable space we have x"— 11‘“ x‘“ =x,. Therefore

the elements y2=yt+¢,(x‘)e Y, at: 1, 2,

form a thread y°e )7. Since

the operator ‘1’, annihilates the subspace Im (15, we have the equations

Ilt.y3=¢ay:‘=¢.J/.=2., whence I} y°=z. This proves that u} is an algebraic epimorphism. It remains to show that III is a homomorphism. For an arbitrary neighborhood of zero V in l7 we must find a neighborhood of zero W in Z such that WV): W. By the definition of the topology in Y, for some y there exists a neighborhood V, in Y7 such that V contains the pre-image of 21/7. For simplicity we suppose that y: 1. Since the mapping W3“: 23—» Y2 is continuous, there exists a neighborhood of zero W: in Z3 such that for arbitrary z3eW3 we can find an element yierp; 1 23 such that y’1=j§ yze V1. So if W is the pre-image of W3, zeW and 23 is the element ofthe thread 2, then the element y, which appears in the preceding argument can be chosen in V,. From what we have proved it follows that

Pi(xl,0)§£-

y?=¥i+¢;(X‘).

If a is sufficiently small, than ¢1(x‘)e V1, consequently y‘eV, and y°eV. I] § 2. Functional spaces 1°. Spaces of functions of finite smoothness Let f be a function measurable in R". Its support is the smallest set supp f: R" such that for an arbitrary point Eésupp f we can find a neighborhood in which f=0 almost everywhere. It is obvious that a support is always closed. Let [2 be a region 3 in R". We denote by 9(9) the space of all infinitely differentiable functions defined on R” and having compact supports lying in 9. We shall soon endow this space with a topology. For every non-negative integer q we consider in the space 9 (R') the following scalar product:

‘= 2

IWD’tNé,

LIISAI R”

all

D’

(1.2)

= 6.59 aggr'

8 By I region we shall mean an arbitrary open seL not necessarily connected.

52. Functional spaces

219

We denote the corresponding norm by

ll¢ll“=l"I*=IHZ IID’¢|li,m~.I*~ .i $4

The completion of 9(R”) in the norm 1] ~||' will be denoted by J“. The norm || ~ II“ and the corresponding scalar product ( -, .>a can be extended to this completion. This makes 6’" a Hilbert space. The most important of the scalar products (2) is (~, ~>°, which we will denote by ( -, ~). Let us fix the integer q go. For any function ¢e9(R“) we define a linear continuous functional on d" by the formula

¢a(¢.¢)=R{ 43:11:15,

11/66“.

This functional is obviously continuous. We have therefore constructed a mapping of 9(R") into the conjugate of 6". This mapping is biunique, since if (4), 1/1) = 0 for all was" it follows that dz 50. We shall show that the image of the mapping 9(R") a (60* is dense in (69?. In fact, if this were not so, the Hahn-Banach theorem would imply the existence of a non-zero element x in the second conjugate space (6‘)” such that (42, 30:0 for all the functions ¢e.@(R"). Since 6' is a Hilbert space, it is reflexive, and therefore the element X can be identified with some function 156'. But if(¢,1)=0 for all ¢e@(R"), we have x20 almost everywhere. Therefore the element xew‘)” is the zero element, and the contradiction we have obtained shows that 9(R") is dense in (69*. When we look on 9(R") as a subspace of (6"? there is an induced topology for it, which is generated by the norm

II¢||“’=SUP{ "4"” was". IIIIIIIHFO}. Illllll' ' We denote by 3‘“ the completion of 9(R") in this norm. Our earlier remarks imply that the space 6’“ is isomorphic to the conjugate of a" and is therefore a Hilbert space. Thus the spaces 6' and 6“ are mutually conjugate. The space 6° obviously coincides, as an l.t.s., with L1(R"), and the space 6"°, being its conjugate, is also isomorphic with L2(R"). This implies the isomorphism 5°36”). These two spaces will be identified with each other. We have thus defined a Hilbert space 6" for every integer q; for all q the space 6'“ is a subspace of J“ (in the algebraic sense), and the identity mapping 6““ —> m is a continuous biunique operation.

220

V. Linear Spaces and Distributions

The support of a function the” where —oo 6" can be continuously extended to a mapping of the factor-spaces

em: n+‘=m+‘/%+‘~a=w%. We note that the mapping eH1 is biunique. In fact, if some function 4556‘“ when imbedded in 6" falls in the subspace 93, its support belongs to G and therefore the function itself belongs to 9 *‘. We remark that since by Proposition 2 the spaces 9} and 6,?“ are mutually conjugate, the mappings d“, and e_,, are also mutually conjugate. The ensemble of the spaces 9: and the mappings d4 forms a decreasing family of spaces. The spaces 61: and the mappings eq also form a decreasing family. We write

9r= lifl (9?, dq)’ than

(32)

£p= (fifllé’l. eq)‘ II-mu

Since 9} and 6’} are Hilbert spaces and are therefore Frcchet spaces, the projective limits (3) are also f—spaces, by Proposition 2, §1. Let us consider the space 95 in more detail. Since all the mappings (L, are biunique, the mappings d": 91,—» 9} are also biunique. Therefore the projective limit 9,- may be identified with the intersection {)9}. The 0

functions belonging to 9}, q; v have continuous derivatives up to order q— v and vanish outside F. Hence it follows that the elements of 9,- are

222

V. Linear Spaces and Distributions

infinitely differentiable functions in R", with supports lying in F. Conversely, if a function ()5 is infinitely differentiable in R", if supp :1) CF, and if all its derivatives D1113 belong to L2(R”), then «1569;. If F is compact, the conditions 0’ ¢EL2(R") may be omitted, since they follow from the continuity of Di 4:. By the definition of the topology in the projective limit, the neighborhoods of zero in the space 9, are sets of the form 9m U, where U is a neighborhood of zero in one of the spaces 91. It follows that the sets of the form ((1): ||¢||" gs} constitute a fundamental system of neighborhoods of zero in 9,. Let us now describe the space 6,. It is clear that the space 6’," consists of all infinitely differentiable functions in R" whose derivatives all belong to L1(R"). For every integer q we consider the exact sequence of Hilbert spaces 0 —> 9?; —> d” —> 6} —+ 0. Since the set F is admissible, the region R"\F is also admissible, and therefore the space 9(R"\F) is dense in each of the spaces 95 . Therefore, for arbitrary q the subspace 93+ 1 is dense in 93. Then Proposition 11, § 1 implies that the sequence of projective limits

0—.9G_.5—»g,,_»0 is also exact. Hence trad/96. We shall show that the spaces conjugate to the spaces (3) admit the following representation:

grammar.charmer-0.2-.» 5" EEWE)‘, 2.21} EM; {gr—‘41-.)-

(4-2)

Since all the spaces 9} and of} are Hilbert spaces, they are reflexive Banach spaces. Therefore the formulae (4) follow from Proposition 4, § 1, which also implies that the spaces 9r and «32 are reflexive. We note that the spaces 9; and 6,? are complete, since they are conjugates of 9'spaces. Sinoe all the mappings e_, are biunique, the inductive limit li_>m («SF—‘1, e_q} is identical to the union U 6;“. When F is compact, the 1

elements of 9‘=U6;4 are called distributions on it. For every distribution f59; the quantity

dezpf=infiqi feé'p“, —°0 MED/5(1), G).

(8.2)

Let us construct the inverse mapping. We choose a function heQG?) which is equal to unity in some neighborhood of the compact K. For an arbitrary function f56(0) the product hf has a compact support and therefore can be looked on as an element of 6. We thus obtain a con-

§2. Functional spaces

225

tinuous mapping 5(9)»! which does not enlarge the support and therefore maps the subspace 6(1), G) into 96. The associated mapping of the factor-spaces is the inverse of the mapping (8). This proves that (8) is an isomorphism. Since the compact K is admissible, the left side of (8) is isomorphic to 6‘, by what we proved in 2". On the other hand, 6(9) is an isflspace and 6(1), G) is a closed subspace of it. Therefore, taking the conjugates of both sides of (8) we obtain on the left space if, and on the right the subspace (of (Q, G))° c 6*(12), by Proposition 6 § 1. This subspace consists ofall those, and only those, 4: e 6* (9) having supports in K. This completes the proof. Let us prove the second assertion. Since 6" (Q) is the space of all distributions with compact supports belonging to Q, the set of its elements coincides with the set of elements in the inductive limit li_'m 6,22,. We need only show that the corresponding topologies coincide. By our first assertion, an arbitrary neighborhood of zero in 6‘ (9) contains a neighborhood of zero in an arbitrary one of the spaces at and therefore in their inductive limit. Conversely, let U be a convex neighbort of zero in h_rn 4!“. Then for arbitrary a the set U.= U at}; is a neighborhood of zero in J;- and therefore contains the polar 82 of some bounded set B. in 6,“. To complete the proof we must find a bounded set B in 5(0) such that 8° C U. We choose a sequence of functions h,cg(fl), which constitute a partition of the unity in 9 and for which:

SUPP haCKa—1\K -3. We set B={2"h,f,feB,, «=1, 2, ...}.

It follows from the properties of h, and B, that this set is bounded in 5(0). Let 4: be an arbitrary element in the polar of B. Since the functions in, form a partition of the unity, we have 4):: h,‘ 43, and this sum is finite. For arbitrary a

|(il.¢,f)l=l(¢,h.f)|§%, feB.. This inequality shows that the function it, 4) belongs to the set 2"c 2" U. Therefore ()5 belongs to the set w

P

UZZ"U¢:U,

q.e.d. [l

fl-l 1

5°. The space 9(0). Again let F and F’DF be closed sets in R". For arbitrary integer q the space 9; contains 91- and the identity imbeddmg

126

V. Linear Spam and Distribution:

91—. 91. is continuous. Since this operation commutes with the operators dq, it can be extended to the limit spaces 9,—>9,. and flat}. Let Q be a region in R” and let (K.} be a strictly increasing sequence ofcompacts tending to 9. The spaces 9‘. and the imbeddings 9K.» 9x. a form an increasing family. We consider its inductive limit

9(0):]39‘,

(9-2)

The space 9 (Q) has the same elements as the union U 9", that is, it is the space of all infinitely differentiable functions in R" having compact supports in [2. Thus the space 9(0) defined by (9) coincides with the space introduced in 1° and denoted by the same symbol. Formula (9) defines a topology in this space; a neighborhood of zero in 9(0) is an arbitrary set containing a convex set U whose intersection with an arbitrary one of the spaces 9‘. is a neighborhood of zero in the latter. In other words there is a fundamental system of neighborhoods of zero in 9(9) consisting of convex sets U, each of which contains for arbitrary or a set of the form

(0359;; ll¢l|'§e, 00 does the set 11 U contain B, i. e. U does not absorb B, which contradicts the assumption that B is a bounded set. Thus BC9K‘ for some a. The sets of the form

V=i¢= ||¢||'§e} constitute a fundamental system of neighbort of zero in 9,“ and are also neighbort of zero in 9(0). The set B, being bounded in 9 (O) is absorbed by each of them and is therefore bounded in 9‘“ I]

67. Functional spaces

227

We obtain the following formula from Proposition 3 {51.

9*(0) = lifl 9;. Using it, we can in particular describe the elements of 9*(0): every continuous functional on 9(a) is characterized by the fact that its restriction on the subspace 92‘, where Kc!) is an arbitrary compact, is a continuous function on 9‘; that is, it is a distribution on K. The quantity degx f may depend on K and is a non-decreasing function of K. The elements of 9" (a) are called distributions on the region Q. If {2’ is a subregion of Q, the identity irnbedding 9(9')—>9(m is defined and continuous. The conjugate operation, from 9"(0) to 9‘6!) will be called the restriction of the distributions in 9 to the sub— region (2’. Every infinitely differentiable function f56(0) and, more generally, every locally summable function in (2, can be looked on as a distribution in 9 defined by the formula

¢->(fl¢)=gf¢d-fThe support of the distribution f59*(0) is the smallest relatively closed subset supp [:0 such that (f, ¢)=0 for all ¢69(n\supp f). The support of the singularities of the function f69‘(fl) is the smallest relatively closed subset sing supp f c9 such that the restriction of f on the subregion 9\sing supp f coincides with some function belonging to 6(Q\sing supp f). Thus the subspace 3(9)::9’ (9) is characterized by the condition sing supp f= Q. Since there exists a continuous imbedding 9(0)» 6(9), every continuous functional on 5(0) can be looked on as an element of the space 9*(9), that is, as a distribution on Q. In each of the spaces 91_ and at there exists, in accordance with 1°, a continuous multiplication by an arbitrary function belonging to 6(9). These multiplications commute with the mappings ea and 11,1, and therefore may be extended to the limit spaces 9(0), 6(0), 9*(0), and (PW). It should be noted that the resulting multiplication by a function f in 6“ ([2) and 9‘62) is conjugate to multiplication by f in the spaces ((0) and 9(0). We introduce one more concept related to distributions on D: degaf= sup(degxf, Kcfl). The quantity deg" f is generally infinite. In 9‘ (9) we single out the subspace consisting of distributions of finite order, i. e. functions for which deg“ f is finite This subspace, with the topology induced by 9"(0), will be denoted by 9“”(0).

228

V. Linear Spaces and Distributions

6°. Bounded sets in 9 (9). Let b = b(n) be an arbitrary positive function of one variable, defined on the ray "go. Suppose further that F is an arbitrary closed set in R". In the space 9,. we single out the subspace 9:: consisting of functions for which the norm

"It“ a: SL119 L MWIID’MI o is finite. The role of the spaces 9} is characterized by the following proposition. Proposition 6. Let Q be an arbitrary region in R". Then I. We have

9(0)=U9'E,

(112)

where the union is taken over all compacts K :0 and allfitnctions b of the type defined above. II. The sets of the form

B=l¢29"x.u¢|1"§1)

(12.2)

constitute a fimdamental system of bounded sets in 9(0). Proofl We first prove Eq.(ll). The inclusion 3 is obvious. Let us prove the converse. Let 4% be an arbitrary function in 9(9). By definition, the set K = supp 45 is compact and belongs to £2. Therefore 45 belongs to the space 9}, where

Mobil}??? IID’ ¢|l°. The function 17 is obviously positive if ¢$0, Let us prove the second assertion. We first show that every set of the form (12) is bounded. As we noted in 5", every neighborhood of zero in 9(0) contains a set of the form (10) for arbitrary at. If a is large enough, K CK, and therefore the set (10) absorbs the set (12), since the functions in (12) belong to 9} and are uniformly bounded, together with all their derivatives. The set (12) is therefore bounded. Conversely, let B be an arbitrary bounded set in 9(0). Proposition 5 implies that B is contained in one of the spaces 9‘. and is bounded in it. Therefore B is contained in an arbitrary set of the form (10), i. e. the functions in B are uniformly bounded, together with all their derivatives. Therefore the function

bo(n)=maxsup(||D’ ¢II°. ¢eBl lllév is finite for all 11 go. Hence it is clear that the set E is contained in the set (12) with K=K. and b=bo. [I

£2 Functional spaces

229

We now note the following well-knovm fact. ‘ Proposition 7. For arbitrary numbers e>0 and fl>1 there exists an infinitely diflerentiable function e in R", whose support is contained in the sphere Ifilée, such that le(§)d{= l, and

sup [0’ e(:§)l é CB"I IjIW

(13.2)

for some B > 0. Proposition 8. Let F be the closure ofsame admissible region Q, and let the function I; satisfy the inequality

bow); €83 71'”.

ago, fl> 1,

(142)

for some Bo>0. Then the space 92 is dense in 9:- for any integer q. Proof. By Proposition 1 the space 9(9) is dense in 9%. Therefore it is suflicient to show that an arbitrary function belonging to 9(0) can be approximated in the norm || - II“ by functions belonging to 9,”. Let d) be an arbitrary function belonging to 9(0) and let e be a function satisfying the conditions of Proposition 7, where the constant [3 is taken to be the same as that appearing in the inequality (14) and s: p(supp :11, C9). We consider the sequence of functions e,(§)=a"e(oc 5), a=l, 2, .... The support of each of these is contained in the e-neighborhood of zero, and therefore the support of each of the eonvolutions e, t d) is contained in the set F. We estimate the derivatives of these convolutions:

lD‘(e.*¢)l=lDiea‘¢|§ CANE)” li|"”, where B is the constant introduced in Proposition 7. Since this constant may chosen as near to zero as we please, the constant 018 can be made less than the constant Bo in the inequality (14). Hence it follows that the function cued) belongs to 9'; for arbitrary or. Since the integral over R’I of the function e, is equal to unity, and its support tends to zero as at —> co, the function e, *4) tends uniformly to (p. The relation D‘(e,, t ¢)=ela D‘ ¢ implies that an arbitrary derivative D‘(e, n: ([1) of this function tends uniformly to the derivative D‘ (15. Hence it follows that eflup a (I) in the norm || - H". [I By what we have proved, the image of the continuous mapping 9; 491, is dense in 9} and therefore the conjugate mapping 6;” —> (9% " is binnique. Hence the space f = U 6} can be looked on as a subspace

in (93*. When F is compact, the elements of (92 " not belonging to 9; are called ultra-distributions.

V. Linear Spaces and Distribution:

230

7°. Sheaves of spaces of distributions We recall a known definition.

Definition 1. A presheaf of linear topological spaces in R" is a correspondence d5: Q~ —»¢(n), which to every region QcR" correlates some l.t.s. @(KZ) and to every pair of regions fl’cfl a continuous linear mapping pg’: Mme 11>(Q') which satisfies the following two conditions: (i) for an arbitrary region 9,4132 is an identity mapping; (ii) for any three regions 9’ :52 ct) we have the equation pg p3 =pg The presheaf 45 is called a sheaf if it has the following property: Let U: (U,} be an arbitrary covering of the region 9. Then I. If f is an element of @(9) such that 93 af=0 for arbitrary or, we have f =0; II If there exist elements fucflUfl) such that for arbitrary or and [i the equation pvrn"!f pU-“v'f holds, then there exists an element f645(0) such that pg‘1:f, for arbitrary or. We note that given any covering U of the region I) we can always choose a locally finite subcovering, that is, a subcovering U'={U,’} with the following property: an arbitrary compact K c (2 intersects only a finite number of the regions U,’. The conditions I and 11 need be verified only for the covering U’, and we may therefore always suppose that the covering U that appears in the definition of a sheaf is itself locally finite. We consider the correspondence

(15.2)

I: Q~ -r6’(t2).

For every pair of regions Q’cQ we choose the mapping pg’ to be the restriction on the subregion [2’ of functions defined in [2. The relations (i) and (ii) are obviously satisfied and therefore these mappings, together with the spaces (15), fom'i a presheaf. It is easy to verify that this we sheaf is a sheaf. Similarly, the correspondence

9': o~—.9*(s2) together with the set of restriction mappings fi': 9‘ ([2) —D 9*(9’) forms a presheaf. The assertion that we are about to prove shows that the presheaf is a sheaf. Let U: {U} be some locally finite covering of a region 0. We shall say that (P: {4) (w), p: } is a presheaf on the covering U if the spaces @(w) and the mappings pm , satisfy the conditions (i) and (ii) and are defined for regions in, (0’ having the form

w=9.

l1..,,.._, =U,on~-nu.v,

v=0,12,....

(16.2)

Let 45 be an arbitrary presheaf defined on the covering U. For every integer v 2 0 we consider the l t. s. v(MU) consisting of cochains of order v

§ 2. Functional spam

on the ‘oovering U with coefficients in the spaces MU.” expressrons of the form

4’: 2 (prev... t. Uta n

n Ur“

231

iv); that is

where ¢lo,i..,i,e¢(ljlo,...,ivli

(17-2)

and where the element 4%....» is skew-symmetric with respect to its induces (see the analogous definition in §3, Chapter III). The topology in the space v45(U) is defined via the isomorphism

new);in 1'1 .¢(U.-....... 1.). in which the right side is the topological direct product of Its In this topology a fundamental system of neighborhoods of zero is formed by sets of the form

{4’2 ¢.E Va, 05!}, where )5 is a finite set of vectors a=(io, ,.., iv), and the K, are neighborhoods of zero in the spaces @(LL). We also recall the definition of the coboundary operators 3'. For arbitrary vgo the effect of the operator (9' is defined by the formula 1

V4.1

5V¢=— Z Z(-1Y$a..r,zj.....i..,”1.0""‘Uem (“3-2) "+2 imam J-o where film", is the restriction of the element (15%," on Ummim (i.e. the result of applying the mapping p333). It is easy to see that these operators define continuous mappings 6": 'd>(U)—>"“¢(U), v=0, l, 2, This collection of n‘ g is K‘ ‘ by the n ' ,r' a 6‘1: ¢(Q)—r°d§(U), which carries the element ¢ into the cochain {$11}. Sometimes, for generality of our notation, we shall write ‘11P(U)= 45(0). We arrange the so—constructed mappings in the sequence

o_,¢(g)i‘_‘,°¢(y)l.

_.V¢(U)l>'+1q)(U)—>m. (19.2)

It is semi-exact, since o'er-1:0 for vgo. Proposition 9. We suppose that the presheaf 4’, defined an the locally finite covering U, satisfies the following conditions: A) For an arbitrary region a) (of the form (16)) 45(0)) is a subspace (in the algebraic sense) of the space 9‘00), and for arbitrary regions w’cw the mapping 433' is a restriction on the subregion w’. B) Let {an} be a locally finite covering of the region a). Then the series 2f , where fiedmo) and supp ficw,, converges in Mai), and the

232

V. Linear Spaces and Distributions

mapping

1] Mafia Us} -> Zfaedttw)

(20.2)

is continuous. C) These is defined in 45(0)) a continuous operation of multiplication by an arbitrary fimction belonging to 5(a)). D) For arbitrary regions 0) and o the following condition is satisfied : if f is a distribution belonging to 9* (w) supp f :0 and its restriction on who is an element of (flame), then fe¢(w). Under these conditions, the sequence (19) is exact. Let us verify that the conditions A) — D) are satisfied by the presheaves ti’ and 9*. For the sheaf 6 they are obvious. For the presheaf 9" we need only B) in the proof. By hypothesis the covering {an} which consists of regions of the form (16), is locally finite. Therefore an arbitrary compact xcw intersects only a finite number of the regions 10,-; for definiteness, say these are wt,

, (ow. Therefore the series

(I, ¢)=Z(fi.¢)

(21.2)

converges for an arbitrary function ¢eQx and is a continuous function on 9”, since only the first k terms of the series can be different from zero. Since the compact KCO) was chosen arbitrarily, the functional f is continuous on all the spaces 9,, and therefore on their inductive limit, which is the whole space 9(a)). Therefore f69* (to). Further, let U be a neighborhood of zero in 9*(w). By the definition of the topology in the conjugate space 9*(19) the neighborhood U contains the polar of some set B bounded in 9 (to). By Proposition 5, the set B is contained in one of the spaces 9,, and is bounded in it. Therefore, on the right side of (21) only the first k terms can be different from zero on the set B. We shall suppose that the functionals f1, ...,f,‘ belong to the polar of the set kB (this set is a neighborhood of zero in 9’(w)). Then the functional f belongs to the polar of B and therefore belongs to U. But this proves the continuity of the mapping (20) when ¢=9‘, q. e. d. The exactness of (19) implies in particular that the presheaf 9‘ is a sheaf. Let us turn to the proof of Proposition 9. We construct the set of functions «56(0) satisfying the following conditions: a) for arbitrary i, supp 01,: 1].; b) 211,51 in {2. Such a set of functions will be called an infinitely differentiable partition of the unity in (2 subject to the covering U. Let (:5 be an arbitrary element of the kernel of the mapping 6": Vii) (U) —» "“d>(U), vgo. We fix the indices i1, ..., iv in an arbitrary way and we choose an index io at will. Let 4);“ M,“ be the corresponding coefficient of

§2. Functional space:

233

the cochain 42. Since the function a“, is infinitely differentiable in 9, Condition C) implies that the product (1.0 tin-ow ,vediwim iv) is defined. This product vanishes near BU“ n ”1......m and therefore it can be extended as a distribution in the region U"... [v with support belonging to Uh, W“. This distribution will be denoted by (an, flaw“). Condition D) implies that it belongs to MUHWJV). We now consider the sum

'14.. ...i, = (V + 1) Z (“to ¢io..... l.)’~

(223)

Since the distribution Wound» is obviously skew-symmetric in its indim, we may consider the cochain

III=ZIIIn..... a. Ur...... i.We shall show that the series on the right in (22) converges in 45(0)), where a): Unwan- Sinoe supp (“an (Island-”Yr: (llama), and the regions Mona) form a locally finite covering of the region a), this follows from Condition B), which also implies that the mapping

H ¢(Ua.nw)9(¢.-m.l..a.} 4 W;.,....:.€¢(w) is continuous for arbitrary fixed indices 1,,

, iv. Therefore the product

"Ml/EH d’(ll.,...,a.)9¢ —> t/IEI'I @(Ui,..... tJ'="""P(U)

(23-2)

of these mappings is also continuous. We must show that the mapping (23) is the inverse of the operator 6"“, Le. that 6V‘1z//=¢. We have

av-‘w= z Za.io(—1>’¢u.....z,.......Utn-mvt. (24.2) 1-9,...,l'., [-

j=

It is clear from (18) that the coefficient of the cochain 6’4) corresponding to the region Ui,im....i. is equal to

l [—jgol— 1)j ¢Lio....,lj,...,lv+¢io...l,iv] v+2 when i$i,, j =0, , v. Since by hypothesis 6V¢=0 this expression vanishes, and therefore the inner sum on the right side of (24) is equal to 45‘0””. If however i=ij for some j, 0§j§v, this inner sum is also equal to 4%.... iv Accordingly, the right side of (24) is equal to 45, q.e.d. We have established the exactness of (19) in all terms except the first. We now establish its exactness in the first term, by showing that the operator 6“ is biunique. Suppose that 6“ ¢=0, where 456MB). For

234

V. Linear Spaces and Distributions

every 1' we write ¢,=p§’, d), and we extend the distribution a; q}, in a, setting it equal to zero on Q\U‘ The extended distribution (ati ¢,) belongs to (15(0), by Condition D), and the series Haida) converges in 93(0) by Condition B). Its sum in the space 9’62) is obviously at. We have only to note that all the terms of this series vanish, since by hypothesis ¢i=0 for all i. It follows that ¢=O. D 8". Coshenves of spam of distributions. We now employ a concept dual to the concept of the sheaf.

W: ” ‘ ‘0..flts, “' "in R”, isa DefinitionZ. A fl~-> W0), that t0r every region {ZCR’I pairs some l...ts 9’62), and to every pair of regions (2:17 some continuous mapping 2}}: Wm)» WET), satisfying the conditions (i) eg=l and (ii) 44:6}, for an arbitrary triplet of regions 0:0: 9”. The precosheaf ‘1’ is a cosheaf if the mapping (P: l2~ —» IMCI!) is a sheaf. The deciphering of this latter definition is lefi to the reader. We use only the notion of the precosheaf. We note the following fact: if d): {(1562), 93’} is a presheaf, the set of conjugate spaces and mappings ¢’= (45‘8”, (pg/Y}, is a precosheaf, which we call the conjugate of the presheaf d5. Conversely, if ‘P= { Wm), efi'} is a precosheaf,

= (WM), (833‘) is a presheaf. In particular, we may consider the precosheavec 9: D~ —>Q(m,

5‘: Q~ arm),

(25.2)

conjugate to the presheaves 9’ and 6. By definition, for an arbitrary pair of regions Qcfl’ the mapping 2“,, needed for the construction of the precosheaves (25) is the conjugate of the restriction mapping pg, in the presheaves 9“ and of respectively. Therefore 6?, is precisely the imbed-

ding 9(a) —» 9(0), or 5* (a) —» 5*(9’).

Now let U be a locally finite covering of some region {2. We shall say that on this covering there is defined a precosheaf ‘P if the spaces 9'00) and the mappings eZ' (satisfying the conditions (i) and (ii)) are defined for regions of the form (16). Let ‘1’ be a precosheaf on the covering U. A chain of order v on U with coefficients in this precosheaf is an arbitrary sum of the form (17) in which the coefficients 4%., IIIII ,v belong to the spaces WU,“ W“), are skew-symmetric in their indices, and have only a finite number difl‘erent from zero. The v-th order chains with coefficients in '1’ obviously form a linear space; we denote it by 'P( U) We may assign it a topology originating in the fact that JP(U): 2 Twin .1») lon/2, the finiteness of the left side implies that the function 150:) belongs to L,. If we apply the inverse Fourier transform to I} we arrive at the inequality

|'/I(§)l§ IWIILé c.1125: l’ Illltz= C Illfill'. Combining this with the bound for the norm of the operator (16), we finally obtain

l¢(§o)| =|Djh(§) ¢(€+§o)lg-ol§ C ||h(€) ¢(€+Co)||'§ C1450) ll¢ll1m where |j|§q—v. This establishes the first assertion. The second is an immediate consequence of the first, since S,‘4=(6,")*. I] Remark. Let us suppose that the function I that appears in the hypothesis of this proposition is logarithmically convex. Then by Proposition 2 the Fourier transform operator F is defined on the space 57'. Let us compute the Fourier transform ofthe functional 6i(2: — in). We have

61(c—co>=(6'(é—éo), expo, .- §))=(5i(§—5o)y expo. —i 6))

=01 expo. —i ole-:.,=(iz)'exp(z, i to).

§ 3. Fourier transforms

249

5°. Sequences of spaces S} and 6} Proposition 4, Let I and 1 be two Admissible functions in R’I such that Q,=n,=R" and I=o(J) as |§|—>oo. Then for arbitrary integer q the imbeddings

SS“->Si.

5}“a

(173)

are compact mappings.

Proof. Let us first establish the compactness ofthe imbedding SS“ —» S} for qgo. Since these are Banach spaces it suffices to show that we may select from an arbitrary sequence @583”), bounded in the norm

ll¢.||3“= Z IIJD’FILI2 (if, ”la“

(183)

a subsequence that converges in the norm ||.||1. By Proposition 3 (52, for an arbitrary compact K the imbedding 6g“ 46,? is compact. Since the norm (18) majorizes the norm 11 ~ Hg“ we can select from the sequence {41,} a subsequence converging in the norm || - ll‘k. If we choose some increasing sequence of compacts K tending to R", we may carry out this operation for each of these compacts. Then, selecting the diagonal subsequence, we obtain a subsequence {4);} which converges in any norm || - II}, where K is compact. For an arbitrary compact K we may write

|I¢lll= Z IIID’ttl‘d-H Z IIID’¢|‘dE. [11541

Ills; CK

(193)

Since I is continuous in R", the first term on the right does not exceed cholq. Since the sequence (05;) converges in the semi-norm ll-Ilfi, it converges in the semi-norm defined by the first term in (19). Since the functions o5; are bounded in the norm (18) and l=o(J) as |¢|—>oo, the second term on the right in (19), with ¢=¢;, converges uniformly to zero as K —)R". It follows that the sequence {05;} converges in the norm (19), q. e. d. The mapping 6f+‘—>£} is also compact for qgo, since we may apply the foregoing arguments to it, with J being replaced by I “‘ and I by J ‘ ‘. When q d".

§4. The exponential representation of solutions

259

Chapter Vl

Homogeneous Systems of Equations In §4, we obtain an exponential representation for the solution of homogeneous systems of equations, of general form, as discussed in the introduction. This representation is the foundation of the greater part of the results in this and succeeding chapters. In § 5, we apply the exponential representation to the study of hypoelliptic and partially hypoelliptic operators. In § 6 we establish for the partially hypoelliptic operators a uniqueness theorem for the solution of the generalized Cauchy problem. § 4. The exponential representation of solutions of homogeneous system of equation The ring 9 will now be interpreted as the ring of all polynomials in a vector iD, consisting of n differential operators

. a . a IE’W’IEE,

. l—]/—l.,

which act in the Euclidean space R". We fix an arbitrary 9-mtrix [1; let its size be txs, We consider the corresponding system of differential equations with constant coefficients

ptiD)u=0,

(1.4)l

consisting of t equations in the unknown vector function u with 5 components. In the context of Eq. (1), we shall interpret it as a column of height 5. Let 45 be one of the spaces introduced in §2 and 3, consisting of functions or distributions, defined on some closed set or in a region of the space R". We shall assume that this space is a 9-module. Then the matrix p(i D) defines a 9-mapping p: (JV—r «D‘. The kernel of this mapping, endowed with the topology induced by if, will be denoted by 4),. If the space 45 is not a 9-mcdule, but there is a wider space ‘1’, such that the operator 170' D) acts from f to ‘1", then we shall say that the function ued)’ is a solution of the system (1) in 9', if it belongs to the kernel of this operator, which we shall also denote by (DP. 1 We denote an arbitrary operator with constant coefficients by p(i D), and not by pm) purely for convenience in the application of the Fourier transform.

260

VI. Homogeneous Systems of Equations

1°. The representation in the spaces of smooth functions and of distributions Theorem 1. Let {N‘,d‘(z,D), i=0, ..., I} be a collection of varieties and normal Noetherian operators, associated with the matrix p’, Further, let (1,, — 00 < a < 00} be some decreasing family of logarithmically convex functions, and let (1,) be a family of logarithmically reciprocal functions. Then for arbitrary integer at and q every element ue[6f_]‘ which is a solution of (1) in 9*(9,_) can be written in the form l

W: Z I 41(27 D) (5W1) Ill

(14)

A=0 N“

for an arbitrary ¢e[Sf_j‘j]’. Here [4‘ is a complex additive measure with support lying in the set N”1 n (Rfixfl1.“) and

lll‘lla—A=;5(lzl+1)”_A‘}a—A(Y)lu‘lécllullza

lt=(#°, -n.#')- (3-4)

Conversely, every functional a, given by formula (2) where the measures a‘ have a finite value Hung, is defined and continuous on the space [Si—j? (that is belongs to [fl‘::]‘) and is a solution of (1) in 9*(l2l._,) and

“I‘ll?" [-9f +2.] WW,’+z._...] (6.4) ”RPS d}—’ 4—» [3").]‘/p* [S’Ep ”J’F—t [SAT/P [SL’+"']‘ where the integer y is arbitrary, and a=li+v+ l =7+20+v+ 1. The composition of these mappings will be denoted by E " By hypothesis, the function u belongs to MET, that is, it is continuous on the space [Sffli The fact that u is a solution of (1) in 9 (0,“) means that the functional pu, which is defined and continuous on the space [S,‘“*"‘]' is equal to zero on the subspace [962, )]‘ Since this subspace —,0 that is, the functional u is dense in [S, ”m ‘, it follows that pu=

VL Homogeneous Systems of Equations

262

vanishes on the subspace p“ [SL‘MJE Therefore we may regard u as a continuous functional on the factor-space appearing in the right side of (6). The continuity of the anti-linear mapping E ‘1 implies that we may construct a continuous linear functional 0 on 13,01’, 11}, which acts by the formula (v, tll) = (u E '1 w). We now assume that the numbers 7 appearing in (4) and (6) coincide. Then the composition E“ E is an identity operator, and therefore, for an arbitrary function ¢6[S,j’]’, we have

(—u,¢)=(“,E —‘E¢)= (v E¢)= (0 1145’)

(7-4)

We represent the functional 1) in the form of an integral with some measure. To this end, we recall that the norm in the space xflp’, d} is expressed by the formula:

If‘(Z)| l|f||,=maXNsup M,(z) Let C‘, 2.20, ...,I be the space of continuous vector functions defined on the set 2N‘ n 0,, the number of components of which is equal to the number of elements in the column d‘ for which the norm II F H = sup {|F(z)|, zeN”191} is finite. The mapping 0

l

moans!» (17%)“; 0* is an isometric imbedding of the space at; {p’, d} in the direct sume C‘. According to the Hahn-Banach theorem, the functional 1) can be extended from atflp', d) to the whole space 6 C“ with conservation of the norm. In View of the general theorem2 the functional 1), extended to e C‘, can be written as an integral

(v,(Fo, ....19)=§5Fm, where ‘1‘, i=0, ...,l are certain complex, boundetL additive measures, concentrated in the sets N‘n0,, and Z I lull = “all. Writing 1.

I‘_A A_ M’ u=

_ 1—0, ..., I,

we may express the functional 1: in the following form:

(v,/>=;mzm f={f‘)e.#;(p’,d),

(3.4)

giM1Iu‘I=);,I lutl= "all.

(9.4)

where

2 See, for example, Danforth and Schwartz [1].

M. The exponential representation of solutions

263

Combining (7) and (8), we obtain the representation (2). The inequality (3) follows from (9), and from the continuity of the operator E “. El 2°. Remarks and corollaries. The representation (2) becomes simpler when the space 6,“. consists of sufficiently smooth functions and, to be precise, if at g A + v, where v = [n/2] + 1. If this inequality is satisfied, then by Proposition 3, §3, the space Sit“ contains all the functionals 6(5 —n) with neg,“ A . Let 2 be the unit matrix of order s. We substitute in (2) the matrix ¢=6(€—q) e, that is, we substitute one after another each of the columns of this matrix. We obtain the row

m=z Id‘(z, D) exp(z’ —ir,) ”I.

it“, W< W A NA

(10.4)

The left side of this equation represents the row formed by the components of the vector function 14. Since the vector u is interpreted as a column, this row is the transpose of this column, that is, it is equal to 14’. Therefore the Eq. (10) yields a representation of the solution 14, if the matrices d‘(z, —in) in the right side are replaced by the transposed matrices (d‘(z, —ir1))’. Thus, Eq.(10) shows that the function u in the region 0,.“ is representable as the sum of integrals with certain measures a‘ over the manifold of exponential polynomials (11"(2, —in))’~ exp (2, —i r1), which are solutions of the system (1). When at is arbitrary, the representation (2) can also be given an “exponential” form. To do this, we note that for arbitrary 269““ the functional

4’ -> ”(A D) $12). defined on [Sf_::]’, corresponds to the functions d‘(z, — i C) exp(z, — i 5). Therefore, Eq. (2) can be rewritten as:

(u. ¢)=;Ng(d*(z, —ieexp(z, 4:), or, where the left and rightt sides are to be understood as functionals on the space [S21'1‘]:Let us realize the representation (2) for the spaces of distributions defined on convex compacts. Corollary 1. Let K and K’: 3K be arbitrary convex compacts in R". For an arbitrary integer at, every function ue[¢f,'(.]' which is a solution of (1) in 9r can be expressed in the form (2), for an arbitrary function

“[93, and llull}"=;N[(lZ|+1)"‘ 1x0) IM‘I é C Ilullfp.

(11-4)

264

VI. Homogeneous Systems of Equations

Conversely, eueryfunetianal, defined byformula (2), with afinite {lull}. , belongs to the space [fi‘fi’=[(S:“)*]’ and is a solution of (1) in 9;, and 3 Ilulli_”§Cll#||'i'~ Proof We construct a strictly increasing sequence of convex compacts KM — oo 0. Then,for an arbitrary region QcR" the natural imbedding 6pm) —> 9; (0) (2.5)

is an isomorphism, and for an arbitrary bounded subregian a): cf) the mapping of the restriction

(Am-MAO»

(3-5)

is compact. We recall that in accordance with the notation of §4, (1,02) and 9:62) are subspaces in [607)? and [9*(Q)]‘, consisting of solutions of the system (1.4). Proof of Theorem 1. We shall suppose, to begin with, that the region {I is convex. We prove the first assertion. The continuity of the imbedding (2) is self—evident. Let us show that the inverse mapping is defined and is also continuous. First, we describe the topology in the spaces appearing in (2). As we established in §2, sets of the form {uz ||u||fi§e where K CI) is a compact, q is a positive integer, and e a positive number, form a fundamental system of neighborhoods of zero in the space 6(0) and, therefore, in the spaces 6,,(9). In the space 9: (Q), a fundamental system of neighborhoods of zero is formed by the polars of the bounded sets in [9(9)]’, and as follows from Proposition 6, §2, we need consider only bounded sets of the form

a={¢: Il¢lllr§%}. where K’CQ is a compact, b=b(r]) is a positive function, and e’>0 is some number. Thus, to show that (2) is an isomorphism, we need only

270

VI. Homogeneous Systems of Equations

for arbitrary K, q and a, find K', b, e' such that every element u of the space 9:62) belonging to the polar of the set .43, is infinitely differentiable in t2 and H u H < e. We fix 13>1 to be not less than 1/1. We further choose some strictly increasing sequence ofconvex compacts {Km — 00 < a < co}, tending to £2, and such that K GK), and we choose a strictly increasing sequence of numbers {an —oo 0.

We denote by C“, where q=0, l, the space of functions in R'l which are continuous if (1:0 and have continuous first derivatives if q=1. We endow Co with the topology of compact convergence and C‘ with the topology of compact convergence of functions and their first derivatives. Then G” q=0,1 are f-spaces. The subspace C'c [C‘]‘ consisting of solutions of (1.4) 1n 9*(R") is closed and consequently IS an f-space also. By hypothesis the imbedding maps Cl onto C". It follows from Proposition 7, §1 that this mapping is an isomorphism. Therefore for every 115C}, the inequality max |D’u(0)|S Clslup |u(¢')|

holds with some constants C and R.

(7.5)

§5. Hypoelliptic operators

273

Let 2 =x+ i y be an arbitrary point of the set N. As we noted earlier, the rank of the matrix p(z) is less than 5. Therefore, for the chosen point 2 there exists a non-zero vector as C', orthogonal to all the rows of p(z). It follows that the vector function a exp(z, — i C) is a solution of the system (1.4). Substituting this function in the inequality (7) we obtain

lZ|§C€Xllyl), where the constant C is independent of 2. It is clear from this inequality that as |z| —>oo, so also |y| —>oo. From this we derive the desired inequality y> 0. I]

2°. Dependence of the smoothness of solutions of hypoelliptic systems of equations on their growth at infinity. The following property of entire analytic functions is well-known: If the function u(£) is of bounded growth at infinity, so also are the derivatives Nu“) as |j|—>no. The solutions of homogeneous hypoelliptic systems have a similar property as is shown by the following theorem Theorem 2. Let p be a hypoelliptic operator with index y, let 1(5) be a spherically symmetric logarithmically convex function, and let J(y)= i(|y|) be the logarithmic reciprocal. Then every solution of the system (1.4), defined in the region Q, and satisfying the inequality l“(€)|§CI((1+€)§),

(8.5)

for arbitrary c>0, also satisfies the inequality

|Diu(€)|§ Cblila (%) l((1+e)§) with arbitrary e>0. Here 930,) is the logarithmic reciprocal of i(exp t). Proof. We choose an arbitrary e>0 and some strictly decreasing sequence of positive numbers {81, —oo — so the inequality

Iz/léBGylHll” holds on the variety N for some B>0. The function fin(r)=inf{|y|+|z”|: zeN, lz’|=r}.

(15.5)

§ 5. Hypoelliptic operators

279

has similar properties. If the projection of N on C, is unbounded, this function is defined for sufficiently large r and is equivalent to tho r7 as r—wo. We set f: —00, if filo=0, and we set 33:00, if the projection of N on C, is bounded. For arbitrary y> —-oo, we have the inequality

II’I§B(|y’|+1z"|+1)‘/’,

B>0.

(16.5)

on the variety N. The following theorem defines weakly and strongly hypoelliptic operators.

Theorem 3. The operator p is strongly hypoelliptic in the variables f’, ifand only if i>0. The operator p is weakly hypoelliptie in the variables 5’, if and only if 7 > 0. Proof. We shall establish the sufficiency only. Since infinite differentiability in either the strong or the weak sense is a local property, we have only to show that it holds for solutions of the systems [1.4) in regions of the form Q=QXQ”, where Q’ and [2” are spheres in R;. and

Re” with centers at the origin of coordinates and of radius p, where p is an arbitrary positive number. For an arbitrary a, 0 no is continuous in the norms Hunk and Huang?“ ". Since 6 > 0 is arbitrary, we conclude that 140 is a distribution in a", and the mapping [9* (Q)]’9u —> nos [9* (17’)? is continuous. For arbitrary jeZ’: we consider the functional (up '1’): Di (“s '1’):'i¢'= {5'

which is the restriction of the corresponding derivatives of the distribution u for 5’ = 60. This functional is a distribution in 9” and the mapping . . u —> uj is continuous. We remark that the functional u is an integral of its restriction (21), that is, the equation

('4. ¢)=I(u, (Nib, 6”))g~|gt=a, diliholds. To prove this, we choose a>0 small enough so that supp ¢ c K.” Then the action of the functional (u, ~)£.1 is given by (20), and the action of u is given by (19), and this yields the result we want. In particular, if (u, -){,._=_(), u=0.

7°. Example: Example 1. Let us suppose that the operator p contains dill‘erentiations with respect only to the variables 5’, that is, piiD)=p(iD¢r), and that the

282

VI. Homogeneous Systems of Equations

operator p(i D§:)—considered as an operator in the space Rgv—is hypoelliptic. So on the variety N’c C, associated with the matrix p’(r’), the inequality |z’|§B(|y’|+l)1h holds. If we look at the matrix p’ in the space R", the associated variety is N’x Cir. In this variety, we have the inequality |z'| g B(|yl + I)”, and accordingly, the operator p in the space 2 is strongly hypoelliptic with respect to the variables 6’.

Example 2. Let s=t=m=1, and let 11(2) be an arbitrary polynomial which admits an expansion in powers of 21 x

11(2): 2111(2’') 1‘1"],

I1=deglh

j— n

(225)

in which pD(z”) is a constant different from zero. We shall show that the corresponding operator p(i D) is weakly hypoelliptic in 51. In fact, in the variety N we have the inequality |21|§B(lz”|+l)‘, where

km“ E a i

J

from which we derive the inequality (16) with i=1/d, and this implies the weak hypoeflipticity of the operator p with respect to 5,. Example 3. Suppose again that s=t= l, and let 61

II

62

"(iD) =—— at? g — at} be the wave operator. The variety associated with the matrix p’ is defined . by the equation #:s We write §'=(C,,...,§,), §"=§1 and z‘= Z

(22, ...,z,,), z"=z,. Comparing the real parts in the equation of the variety, we obtain the relation lifilz—|y’|2=]x"|‘—]y"lz from which we may derive the inequality

lx’l’ély’l‘+lz”l’.

(23.5)

This shows that the operator p is weakly hypoellipfic in the variables 5’. We note that this inequality holds for any change of the coordinate system in C2, for which the new axis 21 remains within the cone lx”| = lx'l, provided that we admit a constant multiplier on the left side of (23). Therefore the operator 1; remains weakly hypoelliptic with respect to E’ for an arbitrary rotation in R", that keeps the new axis of 51 within the cone |§”| =IE’I. In particular, every distribution which satisfies the wave equation has a restriction on an arbitrary time-like line. 6 See, for example, Geld and Silov [3].

96. Uniquuiess of solutions of the Cauchy problem

283

8“. Smoothing operators for the solutions of the system (1.4). Let h: 9' —> 9' be an arbitrary 9-matrix. We shall say that an operator h(iD) smooths the solutions of the system (1‘4) if, for any region DCRE, any solution ue[9‘(n)]', compact Kcfl, and any non-negative integer a, we can find an integer H, such that the function h"(i D) u belongs to [5; ‘. The following theorem characterizes all smoothing operators. Theorem 4. The operator In is a smoothing operator if and only if for arbitrary R > 0 |h(z)|—r0 when zeN, |y|§R, lzI—rco.

The proof is left to the reader.

§6. Uniqueness of solutions of the Cauchy problem 1°. Statement of the problem. Let L be an arbitrary linear subspace in R", with dimensionality between 0 and n— 1. We intend to formulate an analog of the Cauchy problem for the system (1.4) with initial conditions given on L. Since we want to include in our considerations the solutions of this system which are distributions, we must require that these solutions have restrictions on L. We therefore choose a system of coordinates in R" such that the subspace L coincides with the coordinate subspace corresponding to the variables £"=(E,,M, ..., 5,), where n—m=dim L. Setting §’=(€1,...,€m), we assume that the operator 1; is weakly hypoelliptic in the 5'. Then as we showed in § 5, any distribution in an arbitrary neighborhood of zero 0 which satisfies (1.4), and also any of its derivatives, has a restriction on fl”=QnL which is a distribu-

tion. Definition. Let 9 be some neighborhood of zero in R", and let é be a subspace of 9* (Q). We shall say that the Cauchy problem for the system (1.4) with initial conditions on L has a unique solution in 4}, if (1.6) D2, u|L=0Vi the conditions ued>,, imply that I450. We have denoted by d", the subspace in [¢]‘, consisting of solutions of (1.4) in 9*(0). What we usually mean by the uniqueness of the Cauchy problem is a special case of our present definition. 1n the usual form of the Cauchy problem we are dealing with a finite number of initial conditions. For example, if the system (1.4) consists of one equation with one unknown function, and m =1, and the subspace L is not characteristic with respect to the operator p, then on the subspace there are given u=deg 11 initial 6l u " — 661,!‘= 0,..., a —1. conditions

VI. Homogeneous Systems of Equations

284

Since the subspace L is non-characten'stic, the operator Pa in the expansion 1! au-J

P=; p;(D¢~) Ti”! is a constant different from zero Consequently the equation p u=0 and i

the conditions ZiéilL=Q i=0. , u— 1 imply that gig-L =0 for all 1. 2°. A theorem on the uniqueness of a solution of the Cauchy problem Theorem Suppose that on the variety N the inequality

lilé C(IZ”|""+|1/I“’+ 1)

(2-6)

holds with 0< yg 1, 0< fi< 1. Then the Cauchy problem with initial conditions given on L has only one solution in the space of distributions defined on the region 9:17n. and belonging to the space [m', where 1=l(€)=1'(§') ("(5") and

,_ Re, a _{(E’: |§’|0. Hence it follows that for an arbitrary 15 [9 (GMT the convolution 1* l/Ik has the same property. We remark that the integral of the function II'Wklz, taken over the complement of an arbitrary neighborhood of zero, tends to zero as k—» 00, while the integral of '1’; over Ry is equal to unity for arbitrary k. Hence it follows that H x uh — Xlli'" a0. Therefore, for arbitraryj the difference Dé (x 1: Wk) — D3 1 = Di x t w, — Di 1 tends to zero in the norm [I ll‘} Consequently the difference X‘l/Ii—X tends to zero in the noun II - ||",,, which is what we were to prove. Therefore, we have established that the function u is equal to zero in the region w’nu. Ify< l, (3) implies that w’=R;v, whence 1450. We consider the case y: 1. Then by hypothesis the domain of definition of u is the band lfi’l 0 is arbitrary we have

“50 over the whole domain of definition. 7 See Gel‘fand and Siiov [2].

[I

57. Solubility of inhomogeneous systems. M-convexity

289

4". Examples ‘ Example 1. Let s = t, m = 1. Then the reduced order of the system (1.4) IS the smallest p0 such that on the variety N, the inequality

IZ’lé C(IZ"I"°+1)is satisfied. If [10>], then the inequality (2) is satisfied, with y: l, and [3 = l/po . Then the theorem we have just proved guarantees the uniqueness of the solution of the Cauchy problem in the space of distributions defined in the band |§’|0 and A" > 0 are arbitrary. G. N. Zolotarev showed in [2] that the exponent Po/(Po—I) is the largest that will guarantee the uniqueness of the solution of the Cauchy problem. It is reasonable to suppose that the exponents 1/1? and 1/): in the general theorem are also the best possible, but this question has not been investigated.

Chapter VII

Inhomogeneous Systems In this chapter we shall take up the solubility of general inhomogeneous systems of equations in an arbitrary region of Euclidian space and we shall also investigate the possibility of approximating the solutions of homogeneous systems by exponential polynomials. In §7, we give an exact and invariant formulation of these problems in the form of relationships between the functional space and the 9-module corresponding to the given system of equations. In §8, we shall prove an important theorem on the solubility of arbitrary inhomogeneous systems in convex regions. In §§9— 12, we shall arrive at a number of necessary and sufficient conditions for the solubility of inhomogeneous systems and for the possibility of approximations in non-convex regions.

§ 7. Solubility of inhomogeneous systems. M-convexity

We again suppose that p is an arbitrary 9-matrix of size txs. We consider the corresponding inhomogeneous system of equations

p(iD)u=W~

(1.7)

VIlr Inhomogeneous Systems

290

Here w is a known vector-function with t components. We are to find the solubility conditions for this system without imposing on the solution u either initial or boundary conditions. Since the matrix p is arbitrary, its rows may admit differential relationships. The solubility of (1) clearly requires that the vector w satisfy these relationships, whatever they may be. Theorem 1 of this section implies that the local solubility of the system (1) is guaranteed by relationships of the form r(lD)p(iD)=0, where 1’69‘. In order to write down all the relationships of this form,

we proceed as follows: We choose a Q—matrix pl such that the sequence

gm LyLy,

(2.7)

is exact, where p’‘ is the transpose of p,. The exactness of the sequence implies that p’p’1=0, whence p1p=0. Therefore, if the system (1) is soluble, we have pl(iD)w=0. (3.7) We shall show that every vector r satisfying the equation r p=0, is a linear combination of rows of the matrix pl with coefficients from .9. In fact, rp=0 implies that p’ 1’ =0. Therefore, the exactness of (2) implies r’sp‘1 9‘2, that is H=p§ F, where F69“. Hence r=F p1, which is what we were to show. Therefore, an arbitrary relation of the form r(i D) w =0, where r p =0, follows from Eq. (3). The task of finding the solubility conditions of the system (1) we now fonnulate as follows: to determine the functional spaces in which the condition (3) is sufficient for the solubility of the system (1) in the same space. Now we give the exact definitions.

1°. M—convexity Definition 1. Let 15 be a space of distributions and also a topological y-module. Further, let M be a finite 9-module and let M*2 -~-ag'*"isg‘k_s...isyL,-91HM_,0

(4_7)

be a free resolution of this module We shall say that the space (module) 4) is M-convex if the sequence 0—.q)p—»4>‘_'.¢‘L,..._,4ykA..pm-_....y

(5.7)

is exact; we shall denote this by Hom(M, 1)). We recall that the exactness of (5) means that it is algebraically exact, and all the mappings in it are homomorphisms. By Proposition 7 of §3, Chapter I, the M-convexity of the space 43 does not depend on the choice of the free resolution M,. The M-convexity of the space 15 implies that for an arbitrary right side w, belonging to the space “",

i=1.2,... (tl=t, to=s).

By Proposition 7 of §3, Chapter 1, these modules depend, up to a .9— isomorphism, only on M and 115. More exactly, the following assertion is true: Let

..._,mui.w_....irgrL.9«_.M—to be another free resolution ofthe module M. Then for arbitrary i = 0, l, 2, .. . there exist almatn'ces f,- and g, such that the mappings II

r-

‘ppi/Pr—r 45"” Tén/"i—l 4’ ' ‘

(7'50=7‘: P0=Ply

associated with them are defined and are isomorphisms. We shall often use the abbreviation (PM for the modules Horn (M, 415). This means that @M is a space obtained by identifying all the spaces (Pp, where Coker p‘ 3M, with the aid of the isomorphisms f0 and £0. By an exponential polynomial in R: we shall mean any finite sum of the form

f(€)=i(€) CXP(Z;, 46),

2160". “56%

(7-7)

I

We denote by E the space of all exponential polynomials in R; with the discrete topology. An arbitrary differential operator with constant coefficients generates a linear operator in the space E. It follows that E is a 9-module.

Definition 2. We suppose that the space é contains E. Then for an arbitrary finite 9-module M, the space EM is a subspace in d)". We shall say that the space ¢ is strongly M-convex with respect to the given finite 9-module M, if it is M~convex and the subspace E" is dense in 41)". Now using the definitions 1 and 2, we can formulate the solubility problem for the system (1) in the following way: The M-canuexr‘ty problem in the wide sense Suppose given a finite y—module M. We are to describe the class of (strongly) M-convex functional spaces

VII. Inhomogeneous Systems

292

For the spaces 9*(0) and ((0) it can be formulated as follows: The restricted M-convexity problem For a given finite y—module M, we are to define a class of regions (2, for which the space 9* (Q)(J(9)) is (strongly) M-convex. Sections §§7- 12 are mainly devoted to the restricted Mconvexity problem. 2°. Examples Example 1. Let s = t, and let the matrix p (2) not be identically singular. It is clear that “(2)20. Therefore the space (I) is M-convex if and only if the sequence 1" L) d” —> 0,

is exact, that is, if the system (1) is soluble in d)” for an arbitrary righthand side wet?‘ and if the mapping (6) is continuous. Example 2. Let d be the exterior differential operator in R". It operates on the k-th order differential form (k=0, 1, 2, ...) according to the formula (1:14:

=

2 “t.,...,ixd§iiAm/‘dgit—"V n"]' vanishing on E,” but not vanishing identically on 45,. Then ¢e[¢‘]’nE2, and therefore, by (12) dzep‘ [@‘T. It follows that ¢ vanishes on 115,. This however is a contradiction and therefore E, is dense in @F. We prove the second proposition. Let 4) be strongly M-convex. If the sequence (5) is exact, then so is the sequence

0—>¢'/¢,—">d>'—"‘>---. By the first assertion of Proposition 8. §1, the sequence of conjugate mappings is algebraically exact. This implies that (13) is exact, provided that we can Show that the space conjugate to (IF/"/¢15,,)"=[‘1”]x “Eu1»

VI]. Inhomogeneous Systems

296

We now prove the third assertion. It follows from the second part of Proposition 8, § 1 that the conditions a), b), c) and the assertion: (P is Mconvex, are equivalent.

We shall show finally that the following three conditions are equivalent: (1) the space «15 is strongly M-convex, (2) the sequence (13) is algebraically exact, and (3) the sequence (13) is exact. The implication (1) => (2) follows from the assertion II, §1. Since [@‘fl‘nEg is a closed subspace in [@‘T, the implication (2) => (3) follows from the fact that conditions b) and c) are equivalent. Finally the implication (3) 2(1) follows from the fact that 6) implies that d) is M-convex and that assertion I is true.

I]

Proposition 2. Let Q be the union of an increasing sequence of spaces Q“, a: 1, 2, .. . , such that each of the spaces 6(9.) is M—convex. We suppose that for arbitrary a the space 5M9” 1) is dense in dual). Then the space 6(0) is also M-carwex and the space Jum) is dense in 6M9.) for arbitrary a. Iffor arbitrary or the space 662,) is strongly M-convex, then 6(9) is also strongly M—convex. Proof The set of spaces 619,) and of restriction mappings M9,“) —> 6(1),) forms a decreasing family of l. t. s. which we denote by map}. For an arbitrary 9-matrix q we may also consider the decreasing family {6q(fl_)} constructed in a like manner. We fix a free resolution (4.7) of the module M and for every i=0, 1, 2, we consider the sequence of decreasing families

0-* Mum.» -> {[«T(0.)]"}-L> (5,, m (9a)} *0,

(157)

where (10 = p. Since for arbitrary at the space {(0,} is M-convex, we find on removing the curly brackets in (15) an exact sequence of l.t.s. Therefore, the sequence of decreasing families (15) is also exact. We shall verify the fact that the conditions of Proposition 11, §1 are

satisfied. Condition a) is satisfied since 5919,) is a closed subspace of the 9-space [6623“]. We shall verify condition b). If i=0, the space 61452:“) is dense in (SW (12,) by hypothesis. However, when i> 0 the fact that the 6(9) are M—convex implies that 6,112,): p,,,[é’(Q,)]"" for all on. But since the space ($‘(QH 1) is clearly dense in 662“), the space pi_ ,[a‘(9,,+,)]"’I is dense in p,»_1[6(§2,)]"". Thus, we are allowed to apply Proposition 11 of § 1 which in turn implies that the sequence of projective limits of the family in (15) is exact. We now determine the projective limits. Since every compact K :9 belongs to some region 12,, we have madman). It follows that was; (90351.19) for an arbitrary 9-matrix q. Then the limiting sequence

§ 7. Solubility of inhomogeneous systems. Mconvexity

297

is given by:

0 —> 6,10)» mm) "A 5,“..(9) —. 0. It is exact for an arbitrary igO, and this implies the M-convexity of the space 6M. The space 6mg), being the projective limit of the spaces 524(9) is (by Proposition 2, §1) dense in everyone of them. Now, suppose that all of the spaces 6(5),) are strongly M~convex. Then 5M(Q.+1) is dense in film“) since of" is dense in each of these spaces. Therefore, the space 5(9) is M—convex. An arbitrary function 1456,,(9) can be approximated by functions in EM in the topology of each of the spaces 6““); and, therefore, in the topology of their projective limit 6M6?) Thus EM is dense in a?" (Q). [1 4°. M-convexity of the space E Proposition 3 The space E is strongly M—convex for any 9-module M. Proof We have only to show that for an arbitrary 9-matrix p and a matrix p1 such that the sequence (2) is exact, the sequence

(16.7)

E‘ —"> E —"‘+ E".

is also exact. Here for convenience we set p= 11(D) and p‘ =pl(D) where

D: (7:?

, 6—2;) . If an exponential polynomial of the form (7)

belongs to the kernel of the mapping pl in (16), and all the points zj appearing in (7) are distinct, then every term on the right side of (7) also belongs to the kernel of 11,. Therefore, to establish the exactness of (16), we need only show that for arbitrary Z e C’1 the sequence

(17.7)

E;—»’ 324'“ E2.

is exact, here E2 is the space of exponential polynomials of the form f(§)xexp(Z, —i 5). We may limit ourselves to the case Z :0, since the general case reduces to the special case by a suitable translation of the matrices p(z) and 171(2). We may, therefore, suppose that in (17) 2:0. We note that E0 is the space of all polynomials in f=(§l, ..., 6") with complex coefficients. We consider the space 5’ of all formal power series with complex coefficients in n variables z=(zl, ...,z,). We set up a duality relation between E0 and 9’ via the bilinear form

(fl¢)=2fi¢n f=Z-{‘T€‘65m

¢=Z¢af€9€

(1&7)

VII. Inhomogeneous Systems

298

Here we look on every element of the space Eo as a linear functional on .9? It is easy to see that every linear functional on .9’ can be written in the form (18), that is, the space E0 is isomorphic with the space conjugate to 3i We shall show that the operator 0‘, which acts in 50, is conjugate to the operation of multiplication by z’ in .9? In fact, for arbitrary fEEO and 4:53: we have

(fa zi ¢)=Zfi(zj ¢)i=zfi ¢i—j=Zf-'+j ¢i=Z(l)i¢f=(li 43),

which is what we were to prove. It follows that the operators in the sequences (17) are conjugate to the operators in the sequence

@49e

(p=p(zt p.=p.(z)).

(19.7)

Since 5” is a flat 9-module, the exactness of the sequence (2) implies the algebraic exactness of (19). Therefore, Proposition 8, § 1, implies that (17) is exact, which is what we were to prove.

[1

We now obtain a characterization of the spaces 15,. which will be useful to us later. Proposition 4. Let p be an arbitrary 9-matrix of size txs, and let 9(z)=2 6‘ 91(2, 5) be a p’-operator. Then for arbitrary z andj the exponential polynomial 9,12, —i§) exp(z, -i€) (20.7) belongs to the space 15,, and the set of all such polynomials (with the exception of the polynomial that vanishes identically) forms a basis in this space. Here 9} (z, i 5) denotes the matrix obtained from the transpose of D}(z, 6) by replacing the funetionals 6“) by the monomials {—i E)‘/at!. Proof. It follows from the nature of a p’—operator that for an arbitrary fixed 2 the function (20) fails to vanish identically only when

1621\J(2). where 1(2) is the basis set at the given point z, and the functions (20) with jeZ"+\J(z) are linearly independent. It follows that the set of all functions (20) with zeN(p), jeZ’;\J(z) are linearly independent. It therefore remains only to prove that an arbitrary element of the space E, is a linear combination of functions of the type (20) and that these functions themselves belong to [2,. An arbitrary element of the space E‘ will be written in the form f=2f,(—i§)exp(z,, —i{) where all the z. are distinct, and f,(—i§) are the columns of height s, formed from polynomials in —i C. The function f will belong to E, if and only if for an arbitrary function

98. M—eonvexity in convex regions

299

$6 [9 (R")]’ the scalar product (p(I‘ D)L (b) is equal to zero. This equation can be written as follows:

0=(p(x'D)f,¢)=(;m—ic)exp(z.,—i¢),p‘(-‘D)¢) =);(exp0 is arbitrary), and the plane {3 =0, there exists no solution of (9) belonging to 9‘62) which is infinitely differentiable in the neighborhood of 60. Let us suppose the opposite. Let u be such a solution. We note that the operator p is strongly hypoelliptic in the variables §’=(§1, .51), Therefore, on every line §’=const, the restriction function u is defined,

and this restriction depends analytically on §,+i€1. By a theorem of Zerner 3, this implies that the function 14, being infinitely differentiable a priori in the neighborhood of the point 50, is also infinitely differentiable in some neighborhood of the origin of coordinates, with the exception of the origin itself. The existence of such a solution of (9) implies that the operator p is hypoelliptic‘, whereas in fact, it is not. This contradiction proves our assertion. 3 See Zerner [l]. 4 This is easy to show if we begin with the Theorem of the Mean; see §ilov [2}

VII. Inhomogea Systems

308

On the other hand, Corollary3 implies that any distribution in [2 which satisfies the hypoelliptic system (1.7) is infinitely differentiable within K if the right side is so. It is not known, however, whether systems other than hypoelliptic have this property. 4". The role of convexity of the region

Theorem 4 I. In order that the space 9* (Q) or ((9) be M-convex for an arbitrary finite 9-module M, it is necessary and sumcient that every connected component of the region {I be convex. II. In order that the space 9*(9) or 6(9) be strongly M—convex for an arbitrary finite .flmodule M, it is necessary and suflicient that the region Q be convex. Proof. For an arbitrary region wCR”, we denote by @(w) the space 9*(w) or 6(a)). Let {2: UK), be a decomposition of the region {2 into connected components. Then the space 45(9) is topologically isomorphic to the direct product of the spaces 4562,). If each of the regions 9, is convex, then each of the spaces @(flu) is M—convex, that is, the sequence (5.7) is exact with d>=¢(Q,). Accordingly, the sequence (5.7) is exact for the space di=d§(fl), also. But this proves the M—convexity of the space

om).

We shall now prove that the condition is necessary: every component Q, is convex. We suppose that some one of the components is not convex. In this case, we can find a sequence of equal and parallel segments 1., k = 1, 2, , belonging to the domain 9, and tending to some segment I, which belongs to Q, with the exception ofits midpoint. Choosing the system of coordinates in R" in a suitable way, we locate the midpoint of the segment I at the origin of coordinates and its endpoints at the points 5* =(il, 0, , 0). We denote by Q, k=1, 2, ..., the midpoints of the segments 1*. We now consider the differential operator p(iD)=%. We shall 1

show that for some function weaflfl), there exists no solution of (1.7) belonging to 9* (9). Since p,=0, this will complete the proof of the first assertion of the theorem. For every point 6,, we choose the neighbor— hood U, to be such that the sum U,+l,‘ belongs to £2, and the sequence (Uk) tends to the origin of coordinates. For arbitrary k, we choose the function ¢.69(U,,), satisfying the conditions ago and l¢n d§=1. We write ¢k1:; ll¢llk 4’» 5 The norm || . II’ was defined in 5 2.

_ k—1,2,..t5.

§8. M—oonvexity in convex regions

309

Since the regions 6* + U, belong for large k to some compact K :0 the functions ¢;‘(§ —€1) form a bounded set in 9(9). ‘ Let w be some infinitely differentiable function in a, and let u be a disféibution which satisfies (1.7) in {2. The Newton-Leibnitz formula we 5 l

(u,¢l(i—€*))-(u,¢L(€—f'))=f {W(t§++'l)dt¢l('l)d’7‘ (10-8) Since the functions ¢;(.:—§*) form a bounded set, the left side of this equation is bounded. On the other hand, the region U," which belongs to the domain of integration of the right side, tends to the origin of coordinates, which is a limit point of 9. Therefore, the function wean) can be chosen so that l

tw(tc*+n)dz>kll¢ll*.

well..-

Since {45; dr]=l/|]¢||", the right side of (10) is not less than k. But this contradicts the fact that the left side of (10) is bounded. Let us turn to the second assertion of the theorem. The sufficiency of the condition that [2 be convex follows from Theoremsl and 3. We shall prove the necessity. Because of the first assertion, it is sufficient to

show that Q is connected. Let us assume that it is not connected. Let p =d0 (see Example 2, 2°, (5 7). The solutions of the corresponding homogeneous system (1.4) in n are functions that are constant on each of the connected components of [2. On the other hand, the space E, consists of functions that are constant throughout the whole of R". It is clear that one cannot approximate a solution of the system (1.4) using such functions, if the solution is equal to unity on one of the connected components of Q, and to zero on the remainder. This proves the second assertion of the theorem.

I]

5°. Corollarios concerning the solutions of homogeneous systems. The space Q, as a function of the module Coker pf Corollary 4. If the region Q is convex, the spaces 6(0), 9*”(9) and 9"(9) are injective, and the space 6*(0) and the space 9(0), with the discrete topology, are flat 9-modules 6. Proof. It follows from Theorems] and 3 that if the sequence (2.7) is exact, then so is the sequence g): _P_, 4;! _r-, qr),

where 45 is an arbitrary one of the spaces 6(0), 9"”(0), 9‘62). By Proposition 9, § 3, Chapter I, this means that the module 45 is injective. 6 It can be shown that the space 9(0) is a flat 9-module also, with its own topology.

310

VII. Inhomogeneous Systems

Proposition 8, § 1 implies that the sequence

or: i. 4v 4‘. «V is algebraically exact for @= 9(0) and exact for di=l’({2). This proves the second assertion. [l Corollary 5. Let M=.?‘/p’9' and M,=.?"/p;?", i=1, , k, be finite 9-modules and let 4),: M —’M,- be mappings associated with certain 9—mtrices f,: 9‘—r¢‘, i=1, ...,k, such that nKer ¢,=O. Then for an arbitrary convex region (2, every element of 9:62) can be written in the form if,“ D) a}, where 14,5939). Proofl We consider the sequence

OaM—‘AGBMP where the mapping 45 carries the element x into (¢1(x), ...,¢,,(x)). It follows from the condition nKer ¢j=0, that the sequence is exact. Applying to it the functor Hom(-, 9‘19», we obtain the sequence

eo;(o)—l.9;(o)—.o, 1

in which the mapping f carries the element (u,, , a.) into 2 fj’(i D) 14,. By Corollary 4, this sequence is also exact, which proves the assertion. I]

The considerations given in 1°, §7, show in particular that for an arbitrary 9-module (P, the module 45,, consisting of the solutions of the homogeneous system (1.4) that belong to 4", depends essentially only on the module Coker p’t If the 9-matrices p and n are such that Coker p’; Coker n’, then as we observed in 1", § 7, there exist 9-matrices ID and go such that the associated mappings fl): 4),, 4» 45,, establish an isomorphism between these modules. It follows that an arbitrary property of the module 45,, which is conserved under the application of a 9—matrix, is common among all modules (D, with Coker n’ ; Coker p'. We shall often use the notation 4)“ for these modules, where M = Coker p’. In particular, the fact that Q, consists of infinitely differentiable (analytic) functions is conserved under 9~isomorphisms and therefore depends only on the module Coker p’. In consequence, the following definition is correct.

Definition 3. A finite 9-module M is said to be hypoelliptic (elliptic) if it is equal to Coker p’, where p is a hypoelliptic (elliptic) operator; in other words, if for an arbitrary region Q, the space 9:07) consists only of infinitely differentiable (analytic) functions. In view of the results of § 5 M is hypoelliptic ifand only ifthe associated variety N is hypoalliptic.

§8. M-oonvexity in convex regions

311

This definition is useful, in particular, for the solution of the following problem. Suppose given a 9-matrix 42 9'»?‘. The operator p is said to be hypoelliptic (elliptic) with respect to the operator q. if for an arbitrary region (2, every function 1459;“2) has the property that q(t‘D)u is infinitely differentiable (analytic). We now reformulate the definition so that it is invariant with respect to the module M =Coker p’. The mapping q’: .9"—>?fl defined by the transposed matrix q’, is extended to the mapping:

(1’:

?'—>M.

(11.8)

Applying the functor Hom(-,9‘(Q)), we obtain the mapping

9M9) —"> [9“(QlT-

(12-8)

Our definition can now be formulated as follows: we shall say that the mapping (11) is hypoelliptic (elliptic) if the image of the mapping (12) consists only of infinitely differentiable (analytic) functions. It is easy to see that this property of the mapping (11) is equivalent to saying that the operator p is hypoelliptic (elliptic) with respect to q.

Corollary 6. The mapping (1]) is hypoelliptic (elliptic) if and only if its image is a hypaelliptic (elliptic) module. Proofl We construct the f-matrix r so that the sequence

WLWAM

(13.8)

is exact. In the definition of relatively hypoelliptic (elliptic) operators, we may limit ourselves to convex regions. Let Q be an arbitrary convex region. By Corollary 4, the module 9‘62) is injective, and therefore, applying the functor Hom(-, 9'19» to (13), we again obtain the exact sequence

9172(9) 4* [9‘(Q)]‘ —'> [9'(Q)]"Because it is exact, the image of the mapping (12) coincides with 9: ([2). Therefore the mapping (11) is hypoelliptic (elliptic) if and only if the operator r is hypoelliptic (elliptic). However, as we saw earlier, this property of the operator r is really a property of the module Coker r’. Since (13) is exact, we have the isomorphism Coker r’ = Coim q’ '5 Im q’. Therefore, the hypoellipticity (ellipticity) of the operator r is equivalent to the hypoellipticity of the image of the mapping (1 1). [] Let us now consider a problem connected with the local properties of the solutions of homogeneous undetermined systems. We shall say that the operator p and the module M =Coker p’ are virtually hypoelliptic, if every distribution satisfying the system (1.4) in the neighborhood of zero in {2, is the sum of a solution defined in 0 and infinitely

312

VI]. Inhomogeneous Systems

dilferentiable near zero, and a solution with support contained in an

solution arbitrarily small neighborhood of zero. If dim M < u, there is no

of the system ( 1.4) with compact non-empty support (see§ 14). Therefore, the assertion that M is a virtually hypoelliptic module of dimension less than n is equivalent to saying that M is a hypoelliptic module. Let us characterize the virtually hypoelliptic modules. Let M be an arbitrary module. We find a reduced primary decomposition of its null submodules. Let M0 be an n-dimensional component of this decomposition. Then the module MD does not depend on the choice of the primary representation, since it may be defined otherwise: it coincides with the set of elements ¢eM. having the property that f¢=0, for some poly— nomial f69, which is not zero. Corollary 7. The module M is virtually hypaelliptic if and only if the module M0 is hypaelliptic; in other words, if and only if the union of the varieties N‘, associated with M and distinct from C", is hypoelliptic. Proof. We choose the 9-matrix q so that the sequence

y—Lf—‘Ufl.

is exact. Proposition 1, §4, Chapter IV, implies that the kernel of the mapping q’: 9‘4?’ is an n-dimensional primary component p0 of the submodule p’?‘c9“. The submodule po/p’9'cM coincides with Mo and, therefore, we have the exact sequence

0—»Mo—vM-Lv9'. Let 12 be a convex neighborhood of zero. Since the module 9’62) is injective its exactness implies the exactness of the sequence

[9* (9)]’ —"* 9mm ., 939(9) —» 0. Hence every element of 9,: (9) which is a solution of (1.4) coincides, up to a distribution of the form q u, where ve[9*(fl)]', with an element of 9311(0) which is also a solution of (1.4). Every distribution of the form q u can be represented as the sum ofa distribution of the same type vanishing in the neighborhood of zero and a distribution with support in an arbitrarily small neighborhood of zero. It follows that every element of 931(0) can be made infinitely differentiable in the neighborhood of zero by the addition of an element of the same space with a compact support contained in the given neighborhood of zero, if and only if the elements of 9,3062) have the same property. In other words, the modules M and M0 may be virtually hypoelliptic only simultaneously. But since dim M0 < n, the virtual hypoellipticity of the module M is equivalent to the hypoellipticity of M0, and this is what we were to prove. I]

N. Mconvexity in convex regions

313

6". Examples

Example 1. In Corollary 6 we set s: t: r: 1. We determine the conditions under which the image of the mapping (11) is a hypoelliptic module. Since this image is isomorphic to Coim 4', it is hypoelliptic if and only if Ker q’=p0 9, where po is a hypoelliptic operator. On the other hand, as is easily seen, p0 is the product of all the factors of the polynomial p that do not divide the polynomial q. Thus, we may assert that the operator p will be hypoelliptic (elliptic), relative to q, if and only if every irreducible divisor of the polynomial p, which is not also a divisor of q, is a hypoelliptic (elliptic) operator. Example 2. In Corollary 5, we set M =9/p?, Mi=9/pi 9. We assume that the polynomial p is divisible by any of the polynomials P:Then the identity mapping fi’: 9’»? is extended to a mapping of the modules 45:3 M—vMi. The condition A Ker (151:0 means that p is the least common multiple of the polynomials pi. By Corollary 5, every solution of the equation pu=0 in the convex region Q can be written in the form of a sum 2 ui, where ui is a solution of the equation pI ui=0 in 9.

7°. M-convexity in the spaces 6’. and S,‘ Theorem 5. Let I = {1,} be an arbitrary decreasing family of logarithmically convex functions satisfying the conditions Q,‘=fl,-=R", for arbitrary 0:, where la is the logarithmic reciprocal of 1,. I. Let p be an arbitrary 9-matrix. Let 9 =2 {1 9,(z, 6), be a p’—operator, let {N‘) be the set of varieties associated with the matrices p’. and let B c C" be a set such that for arbitrary A the intersection B n N1 is nonempty. Then the linear combinations ofexponential polynomials ofthefarm

9}(Z,—i€)exp(z,—i€),

jeZ'L,

258

(14.8)

are dense in the spaces (6,)1, and (ST), .

II. For an arbitraryfinite ?-module the spaces 6, and S,‘ are M~convex. We observe that all the exponential polynomials (l4) belong to Ep (see Proposition 4, §7). Therefore, it follows from the second assertion that Ep is dense in (6,)? and (57),, so that the spaces 6, and S,‘ are strongly M-convex for an arbitrary module M. Proofofthe theorem. We begin with a general observation. The spaces

X _,= 6,1, — co 0. It follows that the function 43 annihilates all the functionals in the space (titlinWe have thus shown that every functional 4) on the space [6; ]‘ which vanishes on the subspace E", vanishes also on the subspace (6:39,. This means that E: is dense in (6:31), in the norm || ~ "7:1. Since (6,), is the intersection of all the spaces ((1),, an arbitrary function in

§8. M-convexity in convex regions

315

(6,), can be approximated by functions from E", in any of the norms H - 07:1, that is, E: is dense in (60,. As we have shown, the space (S1), is the union of the spaces (5:3,. Therefore, any element from this space can be approximated by functions from E: in some one of the norms || ~||';_... Every such norm is stronger than the topology of the space (5,9,, and therefore E"? is dense in (3“,. This proves the second assertion of the theorem. We now pass to the second assertion. Again, suppose that p is an arbitrary 5'-matrix and p1 is a matrix such that the sequence (2.7) is exact. We consider the sequence of families

0-* {(5).} -’ {WET} —"> l( ilk} 90,

(16.8)

where (EM, —oo WI

WT

WI

(3.9)

0—; °¢M(U)—> [°«P(U)]’—” [“WDTA»

l

l

l

l

0—) oM(o)—» [mm—F» [e(t2)]'i»[¢(9)]"H---

l

0

l

0

l

0

l

0

The exactness of all the columns of this diagram, excluding the leftmost, follows from Proposition 9, § 2. We next establish the exactness of all the rows except the bottom one. Let the cochain ¢=Z 41,.“i q A Uive["tP(U)]" belong to the kernel of the operator p]. This means that p] ¢im ”=0 in via n n Um for an arbitrary array of indices i0, iv. Since every region U,“ n ~ ~- n U," is convex, we know by Theorems 1 8 Here and in what follows the symbol (PM) will mean either of the two speoes 9‘02), {(9). However, writing ¢(0)=l(0) in the right ride of (2), we must write LD=£ on the left side. If 0(fl)=9‘(fl) then 9°=9‘. A similar two-valued meaning will be Assigned to all of our succeeding statements

3113

VII. Inhomogeneous Systems

and 3 of§8, that we can find a function whmmkefififllion mn tinny" such that 4%.. “,=p,_l III“. iv and the mapping ¢Pj(l]io. ...,I.)9¢im ...,lv —' 'l’to, ....ifiDNUaa n

n UMP—7K“ ”1—1

(4.9)

is continuous. Since the function d)“,..... o depends by definition skewsymmetrically on its indioes, we may suppose that the function [11%v is also skew-symmetric in i0, , iv. Therefore, the functions WWW“ are the coefficients of some cochain ¢e[v¢(U)]"". The continuity of the mapping (4) for arbitrary i0, .. . , 1’, implies the continuity of the mapping

'¢p,(U)3¢-> IIIEEWWD'H/Ker 11,-“ which is inverse to the operator Pi~1~ That proves our assertion. We now break the diagram into fragments, each of which is a particular case of Diagram (11) of Theorem 1, § 2, Chapter I. Then that theorem implies the isomorphism (2). We establish the second assertion of the theorem. Together with the diagram (3), we shall consider a similar diagram (3') formed from the spaces v115.,(V), ['¢(V)]'I, v; —1,j;0, and the mappings 5V and 1% By hypothesis, the covering Vis inscribed in U and, therefore, the restriction mappings

'¢M(U)—>'4’M(V),

["P(U)]"->F‘P(V)]"-

are defined. These mappings commute with the mappings 6' and p1 of diagrams (3) and (3’). Therefore, the second assertion of the theorem follows from Theorem 2, § 2, Chapter I.

[I

Corollary]. We have the algebraic isomorphism

ExtV(M,d>(K2));H'(Q,£DM).

v=0, 1,2,

(5.9)

where H“(Q, 50") is the cohomalogy space ofthe sheaf90“ in the region (2. Proof By definition, the space H"(9, $17“) is the inductive limit of the spaces H"(U, .90") and the restriction mappings r; on the set of all open coverings of the region 0. Since 9 is a paracompact topological space, we can inscribe a locally finite convex covering in every open covering of 9. Therefore, the locally finite convex coverings form a cofmal subset of the set of all open coverings of 9. Consequently, in view of Proposition 1, §l, H'(£2,9?M) is the inductive limit of the spaces H'(U,.9'7M), taken over the set of all locally finite convex coverings of Q. Assertion II of Theorem 1 implies that for any two locally finite convex coverings U

§9. Sheol‘ of solutions of a homogeneous system

319

and Vof the region 9, such that Vis inscribed in U, the following diagram: F ”'0’; 9°")

Ext'(M, (15(0))

r;

L H"(U. 90..) is defined and commutative. Therefore, in the isomorphism Ext'(M, 15(0)) §H'(U, 5”"). we may pass to the inductive limit with respect to such coverings U. In the limit we obtain the isomorphism (5). El Corollnry 2. Let M be a hypoelliptie module (see 5°, (58). Then the mappings

Ext'(M, 6(9)) —> Ext" (M, 9‘62»,

v=0, l, 2, .. . ,

(6.9)

corresponding to the imbedding 6(Q)—> 9‘ (0), are isomorphisms. The spaces 3(0) and 9‘ (Q) are simultaneously (strongly) M-convex or not. Proof. We shall consider two particular cases of diagram (3) corresponding to the spaces 43(9)=.!(Q) and 93(9):.“2‘ ([2). The imbedding 6(Q)—rg*({2) generates the mapping of the first of these diagrams in the second. Applying Theorem 2, §2, of Chapter I, we obtain the commutative diagram Ext" (M, 9*(Q));H’(U, 9;") Ext’(M, €(0))§HV(U, a“).

(7.9)

It follows from Theorem 1, §5 that the mappings

wvwamw.

v= —1,0, 1.

(8.9)

are isomorphisms. This implies that the right vertical mapping in diagram (7) is an isomorphism. The commutativity then implies that the mapping (6) is also an isomorphism. Let us consider two particular cases of the sequence (1), corresponding to the spaces @(Q)=o'(9) and 9’((2). In View of the fact that the imbeddings (8) are isomorphisms, these two sequences are isomorphic. It follows that the spaces 6(9) and 9" ([2) are M-convex or not, in unison. Since the spaces JMUZ) and 935(0) are isomorphic, the subspace EM is dense or not in these spaces simultaneously. I]

2°. Examples Example 1. Let M =Coker 115. As we proved in Example 2, 2°, §7, ExtV(M, 45(0)) is the cohomology space of differential forms with coefficients in 45(0). On the other hand, MM is the sheaf V of locally constant

320

VII. Inhomogeneous Systems

functions. Therefore the isomorphism of Corollary] can be written as: Ext“(M, @(fl))§H‘(Q,€). This isomorphism is a particular case of a theorem of de Rham, which refers to a region (2 in Euclidian space. Example 2. Let n=2m and M =Coker”d}J (see Example 3, 2°, §7).

For an arbitrary region 52, the space 45,,(52) is the space 17(9) of functions that are analytic in [2. Accordingly, WM=9£€ where .9? is the sheaf of analytic functions. It follows from Corollary] that we have the isomorphism ExtV(M, 37(9));1-IWQ, 3?). If ¢(Q)=J(Q), it is a particular case of a theorem of Dolbeault. Thus, by Proposition 1. §7, for M-con-

vexity of the space 6(9), it is necessary and sufficient that the equations HV(Q,Jt’)=0, vgl, be satisfied. Since the operator ”do is elliptic, the module M is also elliptic. Therefore by Corollary 2, these equations are necessary and sufficient for the M-oonvexity of the space Q‘m). On the other hand, as is known 9, the equations HV(Q, #)=0, v21, are necessary and sufficient in order that Q be a domain of holomorphy. Thus, M-convexity of the space 6(9) or 9“ (fl) is equivalent to the statement that Q is a domain of holomorphy. As we showed in 4", § 4, the collection of varieties associated with M

consists of one variety containing the origin of coordinates. Therefore, by the remark in 7°, § 8, the condition: EM is dense in Ilium). is equivalent to the condition: the space Eff, is dense in ¢M(Q), where B is the set consisting only of the origin of coordinates. As is easily seen. Elf, is the space of all polynomials in the variables Cj=§j+i Cm +1.. Thus, the condition: EM is dense in @Mm), is equivalent to the statement that the polynomials in (I. are dense in the space #(9) (in the topology of ¢M(Q), which coincides with the natural topology of $62)). This latter condition means that Q is a Runge domain of the first type. Thus, we conclude that strong M-convexity of (15(9) is equivalent to the condition that Q be a Runge domain. 3°. Convexity relative to a zero-dimensional module. Let M be a zerodirnensional 9-module. Then by Proposition 4, §4, Chapter IV, it has a finite dimension ((M) as a linear space over the field C. Theorem2. Let M be a zero-dimensional finite ?»module. For an arbitrary region [2 and a locally finite convex covering U of this region

I. All the spaces H'(U, 9”“), v20, are separable. II. We have the isomorphisms

H"(52. $77M); [H'(0, 91“".

v20.

where V = 61,0 is the sheaf of locally constant functions. 9 See, for example, Serre [3].

(99)

§9. Shelf of solutions of a homogeneous system

321

Praafi By Proposition 4, {54, Chapter IV, the variety N(M) is a finite set, and the ensemble of its points z‘, A = 0, .. . , I, is the set of varieties associated with the module M. Suppose, further, that M g?‘/p’ W, and that 9 = Z 6’ 91(2, 6) is the p’-operator. At every point 2‘ only a finite number I, of the operators 91(2", D) are distinct from zero, and the operators that are different from zero are linearly independent. Let d‘(z‘, D) be a column of height IA , made up of these differential operators. The ensemble of the operators d“(z“, D), i=0, , I, is the set of normal Noetherian operators associated with the matrix 11’. Let a) be a convex region. By Corollary 1 of §4, we can write an arbitrary distribution 14 e (Pl, (0)) on an arbitrary compact m: w in the form of a sum

u(6)=§(d‘(z‘. 46))! exp(z‘, 4:) pl

=29;(z‘. —ioexp(z*, 4m}. LA

(109’

where the u" are vectors whose components are arbitrary complex numbers a}. The number of these arbitrary constants is equal to the sumElA, and by Proposition 4, §4, Chapter IV, coincides with the dimension 1 (M). As we observed earlier, for every 1 the differential operators 91(2‘, D) are linearly independent. Therefore, the ensemble of all the ((M) functions 9}(z“, —r'€) exp (2‘, —r‘C) forms a linearly independent set. Then all the numbers a} are uniquely defined (under the hypothesis that the compact x has inner points), and so do not depend on the compact K. Therefore, the representation (10) is valid throughout the region to. Now let a) be an arbitrary region. We choose some convex covering V={V,}, and we write out the representation for the functions ue¢p(w) in every region If, of this covering. If two regions V, and V” intersect, the representations (10) will coincide in their intersection, and therefore, the corresponding coefficients a} will be the same in this intersection. Thus, we obtain the representation (10) for the functions belonging to (15pm), in which the y} are unique functions of the point i, locally constant in the region to, and therefore constant on every connected compact of this region. By @(w) we denote the space of all complex valued locally constant functions in w. Correlating to the function u the vectors formed from the coefficients [4} in the representation (10), we obtain the linear mapping

mm) —> [?(w)]""’.

(11.9)

This mapping is obviously an algebraic isomorphism. The space Wm) will be endowed with the topology of uniform convergence on every

32

V11. Inhomogeneous Systems

compact KC (0. Then it is easy to see that the isomorphism (11) is a topological isomorphism. Let U be the locally finite convex covering of the regiOn [2 fixed in the hypothesis of the theorem. We consider the complex

0—??(Q)——>°@(U)—W>‘¥(U)ir

(12.9)

which is a particular case of the complex (1) for M =Coker 11;). Here “(€(U), where v=0, 1, 2, ..., is the space of cochains (of order v on the covering U with coefficients from the sheaf g) which is topologically a direct product of the spaces @(Ui0 n - .. n Uiv), isomorphic to the complex line C. (The isomorphism @(Uhn n U“); C is valid, since each of the regions Uin n 0 Us, is connected, since it is convex.) The cohomologies of the complex (12) are by definition equal to the spaces HV(U, g), V: l, 2,

We fix v20 and we apply the isomorphism (11) to all the regions of the form a): Uhn n Uiv. Compounding the direct product of these isomorphisms, we obtain the isomorphisms "¢,(U)§["€(U)]”"’ from which we construct the following diagram:

o—»¢,(t2)—»°¢,(U)—”°»‘4>,(U)—”'>~en en 2n

0—4112) —.°rg(u) Amy) .1....

(13.9)

This diagram is commutative, since every linear operation on the coefficients of the cochain ¢e'¢P(U), in particular, the application of the coboundary operator (7, corresponds to the same linear operation on the corresponding coefficients )4}. The commutativity of the diagram (13) implies that the isomorphisms between the terms of the first and second columns define isomorphisms between the corresponding cohomologies

H"(U,99M)§[HV(U,‘€)]”M’,

v=0, 1.2,

(14.9)

Since these isomorphisms are topological, the separability of the space H'(U, .90") is equivalent to the separability of the space H“((1, fl. By definition Hv(U, (K) = ZV/B', where ZV is the kernel of the operator 6V, and B” is the image 016'“ in (12). Therefore, the separability of the space H" (U, ‘6) is in turn equivalent to the closedness of the subspace B“ c "W U). We shall prove that this subspace is closed. Let (V9'°—>M—>0

(19.9)

and

0—»?“£>9"“—>~~~—>9"—>“ sen—H“ mazflo (20.9) are free resolutions of the modules M and L. Tensoring these resolutions one by another, we obtain the commutative diagram 0

0

0

0

0

l l l l l ..,_, Mm. A Mix b, Mn—u _,..._Ib, Mia —>M®L—>0 l l l l l mgym“;Lyonhgmu—xfi...i,glmufi L10 _,0 i

i”

i”

i”

—“+9“"~&s9‘m~1_,..._£,ghm_,

1. I»

_,9*za-1_,

I

i”1’ In

M

(21-9) _,0

59. Sheuf of solutions of a homogeneous system

325

Here the mapping p’,: WWW—>9”! consists of multiplying the vector (f1, , fim), Le?” by the matrix p}, that is, it coincides with the action of the mapping p;®e: .9"“®9‘1—».9“®9‘1 (2 being the identity mapping); similarly, the mapping q} coincides with the mapping e® q}: 9" @951” —v.9“ @9'1. The exactness of the top row of the diagram and the right column follows from the conditions (1 7). The exactness of the remaining rows and columns follows from the fact that all the modules 9‘ are flat. We consider the free modules 9.: e) 9“”, k=0, 1, 2, ..., and we -k H}—

ccustruct the special free resolution of the module M ®L

may”, Lawn—>2 i»%—vM®L—>0,

(22.9)

where the mappings r; act as follows:

and

Vii:

fl+19(fh+1.0s ---,fo,k+1)—’(gt.o’ "agave?“ Aley'",

grj=Pifi+i.j+(— l)lq;‘fi,j+l'

We shall show that this sequence is, in fact, a free resolution”. For an arbitrary vector f2% + 1 the vector r;_, rgf is by definition formed from the polynomials

hi.1=Pi(Pl+i fi+z.1+(—1)l+‘ Q}fi+u+i) +(— 1)‘ q;(plfi+l,!+t +(— 1" ‘11,“ fun) =PlPl+1 fr+zd+ql ‘11“ ft.1+2=0‘

Thus the semi-exactness of (22) is proved. The exactness in the two last terms follows from Proposition 3, §3, Chapter 1. It remains to show that for an arbitrary k>0, the inclusion Ker rim c Im r; is valid. We consider the module

1m lit—1 n 1m Ila/1m «12.: 113,

(23.9)

associated with the module 9““, where qL_‘ and 171, are mappings in (21). In the diagram (21), we distinguish the fragment formed of the modules 5"”! where i +j=k+1, k—l. This fragment has the form of

the diagram (v) of Theorem 1, § 2, Chapter I, and all the rows and columns are exact. Therefore, the arguments of 4°, §2, Chapterl imply that all the modules H} constructed for this fragment are equal to zero. In particular, the modUIe (23) is equal to zero. 10 The reader who is familiar with spectral sequences can deduce this conclusion . 1 I rnv wim I. .1 u from the fact that L , (see, for example, Gndement [1] Theorem 4.8.1).

326

VII. Inhomogeneous System

Now suppose that f =(fh'o, ,fo‘ k) is an arbitrary vector belonging to the kernel of the mapping r,_‘. This condition, in particular, implies that p’0 f1, ,(_ 1 = —q;_1 la, n- Both sides of this equation obviously belong to the module Im q;_1 0 lm p2,. Therefore, the fact that the module (23) is equal to zero implies that q;_l fo',‘=q,’k_l p}, g where gey'“. Hence q;_ 1(f0’ i—P'o g)=0, and therefore the exactness of the rows of the diagram implies that fo,n'l’i) g=q;h, where he?“’*”. We consider a new vectorf’ =(f,"’0, fo’_k)=f—r,;(0, ..., 0, g, h). It is clearthatfof 0 :0. Since f’— felm 7;, the vector f' also belongs to the kernel of r4“. Then it follows that p’0 f1: 1—1:“) [1" Fl+q fé'.=0. Thus the exactness of the columns of the diagram (21) implies that fl" k—1=Pi g’, where g’e 9‘1“". Further, we write

f”=(fk"’o, ...,f5;,)=f'—r.;(0, ...,0,g',0.0)By construction fgfk=fflq=0 and f”—f’eIm n". It follows that p; [17,” 2:0 and so on. Repeating this argument k times, we obtain

finally the vector f=(0,...,0). Thus, f=f—fe1m rg, which is what we were to prove. This establishes the exactness of (22). We now apply to the diagram (21) the functor Hom(~,¢). and we obtain the following commutative diagram

flk+llni>

in it“ so

1

In:

in»

in:

in

i

i

i

119°") Apnfi. own —“. arm-w —»... [454]” A

l

0

—>['1’M]‘k A [4’u]"‘"—'

l

0

l

0

(24.9) 9L5 [¢u]"'—*0

l

0.

We have used the formula Hom(M',¢)§[Hom(M,d5)]*=[45M]". Here Q,‘ is a mapping which multiplies the vector (14,, ..., u“), where uuedtu,

§9. Sheaf of solutions of a homogeneous system

327

by the matrix q,; the mappings p, and qj have a similar effect. We observe that the lowest row of this diagram represents the result of applying the functor Hom(-, ) to (22), we obtain the sequence 0—’¢MaL—’¢o—m"‘pi—n’ "'-’¢k—m’

“-1" "'r

in which (Pk: 6) WW and the mapping r,‘ is defined by: l+j=h

where

rs: d’hBWLm ..., ¢o,x)—’('f’k+i.an ..., ll’o,k+1)e¢k+1w

¢i,j9¢m’a

lpi,}=pi—l ¢i—l.i+(_l)iqj—1 (bu-1-

Since the sequence (22) is a free resolution of the module M ®L, we

have by definition Hom(M®L,¢)=Kerro, Ext"(M®L,d>)=Kerrk/Im rk_,,

k=l,2, ....

(26.9)

Hence, in particular, it follows that the module Horn(M ®L, 0. Then, if the space 15(0) is M-convex, H'm. ¥)=0 for all i >d. If, moreover, the space 45(9) is strongly M-canvex, then ”‘62, ¥)=0.

§9. Sheet of solutions of a homogeneous system

331

Proof We construct first a finite 9-modu1e L, having the following properties:

A) 5(L)§d, B) Tor,(M, L)=0, i; 1, C) the dimension of the module M ®L is equal to zero. The fundamental ring in 9 will now be interpreted as the ring of polynomials in the complex variables z=(z,,

z"). The existence of the module L will

be proved by induction on n. We shall suppose that a similar task for the ring 9’ of all polynomials in the variables 2,, z,,_‘ has already been carried out. We observe that in the case n— 1:0, the task is trivial. We begin the construction of the module L. If 11:0, we may write L=9’. We shall suppose that d>0i The module M will be represented as a factor-module W/p and we fix some reduced primary decomposition of the submodule p: p=ponwpy In the space C", we choose a co— ordinate such that for all A the submodule p, is normally placed and the intersection of the variety N(p ,1) with the subspace zn=0 has dimen— sion lower by one than the dimension of N (13,). We consider the module L,I =.9/z,I 9. The sequence

aqLn—K) is a free resolution of this module. Therefore 6(L,,)= 1, and by Proposition 3, § 3, ChapterI

Tor1(M,L,,);pnz. 972,9,

Tor‘(M,L,,)=0,

i;2.

We shall show that Tor1(M, L)=O also. For this, it is sufficient to

show that an arbitrary element f ep run?“ belongs to z,' 3). We have f =2" gen, where gey‘. We shall fix some 1; if the dimension of the submodule p, is greater than zero, then in view of Proposition 6, §l, Chapter IV, the inclusion relation 2,' gap, implies the inclusion gem, since the submodule p, is normally placed. If, however, p, is zerodimensional, then by hypothesis the associated variety N (19,) does not intersect the subspace Zn=0. Therefore, 2,, gep,1 implies gep,1. Thus, we have shown that gep, for arbitrary A, that is, gep. Hence f:2,' 362,, p, which is what we were to prove. We have now shown that

Tori(M, L,,)=0,

i; 1.

(33.9)

Let us consider the 9’-module m, which is the restriction of the module M on the subspace zn=0 (see 5°, § 1, Chapter IV). By Lemma 2, §1, Chapter IV, the variety N(m) is equal to the intersection of N(M) and the subspace z,I =0 and, therefore, of dimension d —— 1 because of the

choice of a system of coordinates in C". The induction hypothesis implies that there exists a finite 9’-module I, having the following property:

332

VII. lnhomoyneous Systems

a)6(!)§d—-1,b)Tor,(m, 0=0forig 1,andc)dim(m®l)=0. Suppose that 0—.9'u_...._.glrx..

k_,glrk 0: I

a?“—°>9’”°—>I—>0

(34.9)

is a free resolution of the W-module l of length 6 =60). We now replace 9' by 9 in this sequence, and we replace 1 by the 9-modu1e L’ = flw/qo 9“ : 0_,gi,.,._,yrku_, 9n. 4: I

(35.9)

~~~—*9"JL’9‘”—>Ea0.

We show that this sequence is exact Let fe?“ be an element belonging to the kernel of q,‘_1. We expand it in a power series z": f=z fi 2:. The equation q,_ 1 f: 0 implies that q._ 1 f: 0 for arbitrary 1’. Since the 69"” whence f: q, g, that sequence (34) IS exact, we have I: q,g,, gie is, g: 2 2159“". This establishes the exactness of (35). Thus the sequence (35) is a free resolution of the 9-module L’, whence 6 (L’)< d— 1. Let us set L=L‘®L.. Proposition 5, (3‘4, Chapter IV, implies that Tori(L’, L") = 0, i g 1. Therefore, by the remark in 4°, we have the inequality 6(L)§6(L’)+6(Ln)§d. This proves property A). Let (19) be a free resolution of the module M. The relation (33) implies that the sequence

...—>E:LpE;—”"—vE:—»M®Ln—>0.

(36.9)

is exact. We now remark that the operation of multiplication of the ring 9 generates a multiplication operation among the cosets in the factor-module L,,=.9/z,, 9. This operation converts L" into a ring isomorphic to 9’. The 9-modules M ®L,I and L are also Ln-modules. The ring isomorphism Lug? generates an isomorphism of the modules M®Ln§m, Lgl, and the sequence (36) can be looked on as a free

resolution of the ?‘—module m. It therefore follows from b) that the sequence ,

..._.uzi»L'-A» '0.

is exact. If we compare this sequence with the resolution (19), we arrive at the conclusion that Tori(M, L)=0, i; I. This proves property B). Let us prove property C). Proposition 3, § 1, Chapter IV implies that

N(M ®L)=N(M ®(L’®L,.))=N(M)nN(L’)nN(L,I).

(37.9)

The variety N (L,) clearly coincides with the subspace z,I = 0. By Lemma 3, § 1, Chapter IV, the intersections of this subspace with the varieties N(M) and M (L’) are varieties N(m) and N(I). Therefore, the right side of (3 7) is equal to the variety N(m) n N(I) = N(m ® I), which has dimension zero in accordance with c). This proves property C).

§9. Sheet of solutions of a homogeneous system

333

We now proceed to prove the theorem. We shall establish the equation

Ext‘(L, ¢M)=0,

i>d,

¢=d>(t2).

(38.9)

It follows from A) that a suitable free resolution of the module L has a length not exceeding 11, that is, it has the form (20). Let (24) be a diagram constructed with the aid of such a resolution. The bottom row consists of zeroes to the right of the terms [IPMTE Therefore, the spaces (38), which are the cohomology spaces of this row, are equal to zero for i>d. We shall establish Eq. (38) with i=d, on the assumption that the space 1) is strongly M-convex. This equation is equivalent to the following: qd— I [‘DMT'H = [¢M]"~

To prove that the left and right side coincide, it is sufficient to show that a) the left side is closed in the right, and b) that the left side is dense in the right. That the left side is closed is equivalent to saying that the space Ext" (L, (PM) is separable. It follows from Theorem 3 that this space is isomorphic to Ext‘(M ® L, (D). The latter, by Theorem 1, is isomorphic to the space H'(U, mus L, where U is some locally finite convex covering of the region 12‘ It follows from Property C) and Theorem 2 that the space H"(U. 9M9 L) is separable. This proves the assertion a). We prove assertion b). Since by hypothesis the space EM is dense in d(igd).

(39.9)

Since each of the zero-dimensional modules is not equal to zero, it follows that l (M (X) L)+ 0. Therefore our theorem is proved. I] 6". Examples Example 1. Let p be a square 9-matrix of order s, with a non-constant determinant. The variety associated with the module M =9'/p’ 9', is

VII. Inhomogeneous Systems

334

the variety of roots of the polynomials detp (see Proposition 2, §1, Chapter IV), and, therefore, has dimension n— 1. Then, by Theorem 4,

the space (Pu?) will be strongly M-convex only if H"1([2, ¥)=0, that is, only if the complement of Q in R" has no connected compact components. Example 2. Let p and q be elements of the ring .9. We set M =?/q? and L=9/p 9. We shall find the conditions on the polynomials p and q, for which Ton—(M, L)=0, i; 1. By Proposition 3, §3, Chapter I these relationships are equivalent to the equation p 9 n q .9 = p q 9. This equation means that every polynomial divisible by p and q is divisible by the product pq, that is, that the polynomials p and q are relatively prime. We thus suppose that the polynomials p and q are relatively prime. Then Theorem 3 implies the following proposition. Let Q be an arbitrary convex region. Then the equation pu=w, where wedJm) and qw=0, has a solution ue M a0. In both cases, t=0, and therefore, the corresponding system (1.7) is empty. Thus the problem of M-convexity for 6(M) 0,

where p’4:0. Theorem 1. Let 6(M)=1. Then the space 3(0) is M-convex if and only if the following condition is satisfied: (5,) For an arbitrary compact Kc!) there exists a compact K’CQ such that the relation supp p'dJCK, where ¢e[9(9)]‘, implies that supp 45 c K’. Proof Sufficiency. Suppose that (Sn) is satisfied. We shall show that it holds also for functions in [6*(fl)]‘. We fix an arbitrary compact

336

VI]. Inhomogeneous Systems

K c9, and we choose the compact K0 to be such that 02KB: 2K. Let (1% be an arbitrary function belonging to [6' (9)] such that supp p’dz CK. We choose a function 159M“) satisfying [1 d5 =1. The sequence tends to the delta-function in the of functions 1v=v"1(v§), v=1, 2, topology of distributions. For sufficiently large v the support of the convolution x“ t p’¢=p’(xva 45) belongs to K0. The hypothesis (Sn) implies the existence of a compact K', depending only on K0, and such that supp()gv v ¢)c K’. Since xv: ¢a¢ for v —> no, we conclude that supp ¢cK’. But this proves that the condition (Sn) holds for distributions with compact supports. Let K,, at =1, 2, be some strictly increasing sequence of admissible compacts tending to (2, such that for arbitrary a the compact (K; corresponding to K, in (5,) belongs to KM”. The Frechet spaces ‘1. obviously form a decreasing family. For every 11 in ,[d',?_]‘ we consider the subspace fi=[6,f_]‘np‘[6‘(fl)] and its orthogonal complement I; in the conjugate space [6“? The spaces pf form a decreasing sub~ family of the family ([JK‘J‘}; we consider the sequence

0—. (12") H {[JKJ‘} J—e ([JKJ'} —> 0

(1.10)

where the mapping p is generated by the differential operator p. We show that this sequence is exact. It is clear that 1:0 belongs to (6“),5 on the other hand (Sn) implies that 1:: p*[(;m]' (here we make use of Proposition 4, §2) whence we may conclude that the image of the restriction mapping (thug—46“)” belongs to If). This implies the exactness of (1) in the middle term. Let us prove the exactness of (l) in the third term. We fix at and the function heQGntK,+1) equal to unity on K, Choosing wewKHJ' arbitrarily, we extend the product h w as zero outside of K“, The function so obtained belongs to [£(R')]' and therefore by Theorem 3, §8 there exists a function ue[d'(R‘)]‘ satisfying the system pu=hw. The operator wn ulx. (generally, many-valued) acts continuously from MINT to [£K_]'. The family mapping having these operators as components inverts the mapping p. This proves the exactness of (1). We now show that the subspace fl is closed. In view of Corollary4, §8 the mapping

[6‘(R")]‘ L [4"(R')]‘

(2-10)

is a monomorphismt Proposition 4, §2 implies that the subspace [ff-”T is closed in the left side of (2). Consequently its image is closed in the right side. The subspace P. is equal to the intersection of this image with the closed subspace [JET and therefore is closed in the space [f*(R")]‘. By Proposition4, §2 we conclude that I; is closed in [ILJ‘ a so.

910‘ The algebraic conditions for M-oonvexity

337

We shall verify that the sequence (1) satisfies the conditions of Propositionll, §1. The condition 3) follows from the fact that P0 is a closed subspace of the 9-space [51“]:- To prove b) it is sufficient to establish that any functional ¢e[6*]' which vanishes on P + 1 also vanishes on P". Since the space 6* is reflexive and P+1 is closed, the functional d) belongs to P"1 and therefore to P+,n[6’* ]’= P. Hence ()5 15 equal to zero on 1:. Propositionll, §l implies that the sequence

[4"(Q)]' —"—t [619)? -> 0 obtained from (1) by passage to the projective limits, is exact. This completes the proof of sufficiency. Let us now prove the necessity of the condition. We shall suppose that the space 6(9) is M-convex and we fix an arbitrary compact K c9. We consider the space '1’, formed of the distributions ¢e[d"(fl)]’ for which supp p" 45 c K. We shall endow this space with a topology defined by the semi-nouns || p*¢ ||", k=0, 1, 2, . We consider the bilinear form

(ߢ)= [M 11:.

(3.10)

defined for functions fe[d’(fl)]' and distributions 4555". For a fixed distribution (1), this form is continuous in L since :15 has a compact support. 0n the other hand, since the space 6(0) is M-convex, for an arbitrary function f we can find a function ue[d’(fl)]’ such that p u=f.

“mm

(L ¢>=a, EEC, satisfying the following conditions: a) z(£)=§" W+x({) for some natural u,

b) |K(€)| §c(ll|“+ l)‘for some h< 1. We now formulate the concept of a hyperbolic variety. Definition. Let (6‘, 5“) be some division of the variables 6 into two groups, in which the group 45' is non-empty; let (2’, 2”) be the corresponding division of the dual variables. We shall say that the algebraic variety N c C: is hyperbolic in the variables 5’, if for an arbitrary improper point (0, w', w“) belonging to this variety, the condition Im w”=0 implies that Im w’ =0. Let L be a subspace linear in R", distinct from R". 12 For the proof, see Matruura [l].

VII. Inhomogeneous Systems

340

We shall say that the variety N is hyperbolic with respect to L, ifit is hyperbolic with respect to the variables 6’ so chosen that L is the manifold of solutions of the system (=0. Theorem 2. Let Q be a finite or countable union of convex regions U“ satisfying the following condition: for an arbitrary variety N‘, associated with the given module M, we canfind a hyper—subspace L in R", with respect to which the variety N‘1 is not hyperbolic, and such that the projections ofthe regions (L on Ll are pairwise disjoint. Then the space 45(9) is strongly M-convex.

Proof We shall suppose that the variety N(M): u N‘ is empty. Then the module M is equal to zero, and therefore, the theorem is true

in an obvious way. We now suppose that the variety N (M) is non-empty. Then the hypothesis of the theorem is meaningful and it implies that the regions Uv are pairwise disjoint. Therefore, the M—convexity of the space M0) follows from Theorem 4, § 8. It remains to show that EM is dense in 11)" (9). Proposition 1, § 7 implies that for this, we have only to establish the inclusion relation

P“ want: [45* (0)1‘ n E:

(5.10)

(p is a matrix from the resolution (4.7) of the module M). Let 11> be an arbitrary element of the right side of (5). Since the regions U are pairwise disjoint, the function ¢ is uniquely determined as a sum Zip“ where ¢aVe[(l>‘(Uv)]'. We shall show that ¢VEE° for arbitrary v. We fix an arbitrary variety N‘1 associated with M. It follows from the conditions of the theorem that there exists a linear manifold LCR" of dimension n— 1, separating the regions U1 and U'= 2 IL, with respect v> l

to which N‘ is not hyperbolic. In R", we choose a system of coordinates such that the subspace 51:0 coincides with L, and the region Ul(U’) lies in the half--space fl>0(§10, respectively {1 0(y1*(R")]'. It follows that

(7.10)

x*(2)+x'(z)=p‘(2) «5(2)

Since by hypothesis the variety N‘ is not hyperbolic, relative to the subspace L, it has at least one improper point (0,wl,w") such that Im w”=0, but Im w, *0. Let Z(§)=C"(W1,W")+K(C) be an algebraic curve belonging to N‘, constructed with the aid of Proposition 1. Suppose, further, that 9:2n‘9i(z,6) is a p*—operator. We consider the sequence of functions x?(()=9i(z(§), D) x*(z(§)). By Theorem 1, §4, Chapter II, for arbitrary i the coefficients of the functional 9,-(z,6) are rational everywhere regular functions on the set of the form N,,\Nk+1. where (M) is some algebraic stratification. Since the curve z=z(§) is defined and algebraic for |C|>a, it belongs altogether to one of the sets N.‘\N,‘+ 1 for I: | > (1'. Therefore, the coefficients of the operators 9,- (z((), D) are holomorphic and algebraic for |£|>a’ and, therefore, they grow at infinity no faster than some power of ICI. Then for arbitrary i, we have the inequality

|x:*(DI§C(ICI+1l" sup 11*(2)l, Ix—xmlsx

i621.

(8410)

Making use of (6) and b) we obtain the bound

lzflillé C eXp(A'lC|“),

(9.10)

which holds for arbitrary it We estimate the function x5“ for real values of C“. Suppose for definiteness that Im wl>0. We shall assume to begin with that (”>0. From (6), (8), and b), we have

lam-"(Olé C(ICI +1)" SKIN-6' 11'“ W: (“+4 MOI) é C’exp (— % Im wl C“) .

Now suppose (” 2 to Ep, and so on. Thus, we have proved that for arbitrary v,

¢v€ [¢*(Uv)]‘ n E:Since all the regions U" are convex, it follows from §8 that ¢v=p* WV, where xfive[¢‘(Uv)]‘. Setting #1:: 1/1,, we obtain ¢=p*lp6p" [@‘(flfli This proves the inclusion relation (5).

[]

Remark. We can show that when (2 is the union of two non-intersecting convex regions U1 and U2, the condition for strong M—convexity obtained in Theorem 2. is also necessary. In fact, the following assertion is true: In order that the space @(9) be strongly M-convex, it is necessary and sufficient that for arbitrary A the variety N“ be non-hyperbolic with respect to some linear manifold L“ of dimension n—l, separating U1 and U1. When there are more than two regions U” the condition of Theorem 2 is no longer necessary. Suppose for example n=2, and 12 is , 2 41: . the exterior of three rays, 4) =0, Tn, 1n the plane. Then by Corollary 1. 3 the space (15(0) is strongly M-oonvex for an arbitrary elliptic module M. The hypothesis of Theorem 2 is not fulfilled, since no line divides the connected components of the region 9.

3°. Homological conditiom for M—convexity. The conditions in question amount to this: a given finite module L can be included in an exact sequence of the form

0—.La9'~-iiagu-z_t..._vi ,gn

pa

93°.

(10.10)

In §13, we shall find the conditions on L. under which such a sequence can be constructed. We show here that the longer the sequence, that is, the larger the number k, the wider the class of regions {1 for which the sequence 45(9) is (strongly) L-convex.

§lO. The algebraic conditions for M-convexity

343

Theorem 3. we assume that the module L can be included in an exact sequence of the farm (10), of length k, and that the region Q has a locally finite convex covering U consisting of regions, having no more than k+1-fold mutual intersections. Then I. The space 45(9) is L-convex. II. Let 9’ be a region having a convex covering U’ inscribed in U. Then the image of the restriction mapping 115(9)» 45,157) is dense in 4540’). Proof. Suppose that ...Agm._2&_,yu_,L_,o

is a free resolution of the module L. We join it to the sequence (10). To this end, we consider the mapping p;_ 1 , the composition of the mappings 9“ —» L a .9“ 4. It is easy to verify that this mapping makes the sequence

meow—i“ rem—M m--—»---—»P‘ ?'—"">9"’—>M~+0 (11.10) exact. Here we have written M =Coker 111,; thus this sequence is a free resolution of M. Let us consider the following commutative diagram 0

0

0

ill ”l l l l ”l

*mm—M “mm—M" wwfi'u k—1¢(U)

po

k-1¢(U)

p1

Pk-l °¢(U)

i l

n

145(0)

n

0‘15“!)

pm °¢P(U)—>--~

cum—"H «rank? Maw-~°

0 (12.10)

where for simplicity we have omitted the brackets of the form [...]“. It is a particular case of the diagram (3.9) (from which the left column has

344

VI]. Inhomogeneous Systems

been removed), since the hypothesis of our theorem implies that "+ 1€15(U)=0. In the course of the proof of Theorem 1, § 9, we showed that all the columns and all the rows of this diagram except the lowest were exact. Therefore, on the basis of Theorem 1, §2, Chapter I, we conclude that the bottom row of ( 12) is also exact. This proves that 115(1)) is convex. We now prove the second assertion of the theorem. Since the region (2’ belongs to (2, and the covering U’ is inscribed in U, the restriction mappings d}(£2) A» dim’), "D(U)a '¢(U’) are defined. By these mappings the topological spaces 45(9) and "45(U') induce the topologies on the spaces 45(9) and "4)(U). The latter with the induced topology will be denoted by (13(0) and "H U). Let us consider the diagram (12), obtained from diagram (12) when we replace the M9) and ”45(U) by 5(0) and @(U). We shall prove that the space 6(0) is Mvconvex. To this end, we prove that all the rows and columns of the diagram (12) are exact. Let U=(U,.}, U‘={U,—’}. For arbitrary i0, ..., i,,, we denote by MU,“ n n U“) the space (MUin n ~ ~~ n Ur)» in which we have introduced the topology induced by ¢(Ui;n-~-n(1,-;). The space '¢(U) is clearly the direct product of the spaces 5(Uiun-u n Ur.)- Since by hypothesis all the regions U, and U,’ are convex, it follows from Corollary 2, § 8, that each of the spaces @(Uio n n 1],") is M—convex. It then follows that the spaces “5 (U), v20 are also M-convex. In view of the exactness of (11), we have now proved the exactness of all the rows of the diagram (12), except the lowest. The set of spaces (15(0), 5(1)“ n - ~ A n Us) is a presheaf on the covering U. This presheaf, clearly, satisfies the conditions of Proposition 9, {$2. But this proposition implies that in the diagram (12), all the columns are exact. Thus, in diagram (12), Theorem 1, §2, Chapter I, is applicable, and implies that the lowest row is exact, that is, the space (5(0) is M-convex. Since L=Coker p}, we have ¢L(Q’)=d>pk(f?). Suppose that u is an arbitrary element of 43(0). Since the space GHQ) is dense in «15(0), we can find an element oe[¢(9)]"‘, belonging to an arbitrarily small neighborhood of u in the topology of [@(Q')]'*. Since the operator pk is continuous, and pi u=0, the element pk!) is arbitrarily near to zero in the topology of the space [¢(Q)]"‘". The M-convexity of (5(9) implies that the mapping 1),: [43(0)]’“—> [11>(£Z)]"“I is a homomor hism. Therefore, the element 0 is arbitrarily close in the topology of [3253]“ to the sub space ‘15,,(9). Therefore, the element 14 is also arbitrarily close to the subspace 43,49), which is what we were to prove.

[I

We observe that Theoremsl and 3 of § 8 can be looked on as particular cases of the theorem we have just proved, corresponding to the case k=0.

NO. The alybraie conditions for M-oonvexity

345

Theorem 4. We suppose that the conditions of Theorem 3 are satisfied, and we impose the following conditions also: for an arbitrary variety N‘. associated with the module M that appears in (11), there exists a hypersubspace L‘ in R", with respect to which N‘ is not hyperbolic, and such that for arbitrary i1, , i. the projections of the regions Vin=UionUilnaiv

io$il,...,i,,

on (LA)1 are pairwise disjoint. Then the space (15(0) is strongly L-corwex. Proof The L—convexity of dim) follows from Theorem 3. Let us consider the commutative diagram 0

0—»

l

0

l

“@W(U)—>

‘tP(U)-—’°—r"¢>(U)

0

0

a

Laura!)

0~"‘¢n(U)——M-‘¢(U)J—» 3

0

a

MWMWL °d>(U)—""t "@(U)

ill

«NML @(fl)¢> 90(9)

0

0

which is obtained by extending diagram (12) one step to the left. By what we have proved already, all the columns of this diagram are exact, except the leftmost, and all the rows except the lowest. Therefore, we may derive from Theorem 1, (52, Chapter I, the isomorphism

‘4’p(U)/t7'“¢,(U)’=”¢n.(fl)/pn_1[45(9)]‘H,

p=po-

(1310)

We shall show that the factor-space on the left is absolutely inseparable, that is, the only open set in it is the space itself. Let u=2 um n’ U,“ AA Uik be an arbitrary element of the space *¢,(U). We fix arbitrarily the indices i1, .. . , it, and we consider the components alo=uhh ik of the function u. Each of them is defined and satisfies the homogeneous

VIL Inhomogeneous Systems

346

system (1.4) 1n the regionV .Sinoe these regions are pairwise disjoint the function v.0 defines a single-valued function v in the region V= U K”, belonging to the space ¢,(V). The hypothesis of our theorem implies that the covering of the region V, formed by the regions K0, satisfies the conditions of Theorem 2 with respect to the module M. Therefore, by Theorem 2, the space «15,,(V) is strongly M-convex. Therefore the function v can be approximated by exponential polynomials v’eE,. Since the indices 11,.. , 1",, were arbitrarily chosen, we may assert that the cochain ,1]- A /\ U , ucan be approximated by cochains ofthe form 14 = Z v; where all the v- _____ ,ke E .The cochains u ofthis form belong to 5" ‘ d),( U). and therefore, the left factor-space 1n (13) 1s absolutely inseparable. Since the isomorphism (13) is topological, the factor-space on the left is also absolutely inseparable, and therefore, the subspace p._ 1 [@(Q)]"‘" is dense in 451:1 (9). Since the exponential polynomials are dense in 45(9), the exponential polynomials of the form P171 f, fGE'k" are dense 1.11 ph_,[¢(0)]“‘" and, therefore, in (final). The exponential polynomials of this form, obviously, belong to E”. Therefore, the space Epk is dense

in (FAQ).

[]

Corollary 1. Let the region Q admit a locally finite convex covering, consisting of regions that have no more than k+ 1-fold intersections. Then for an arbitrary finite 9-module M Ext‘(M 115(9))=0,

i>k.

(14.10)

If, moreover, the module M satxsf‘es the conditions of Theorem 4, the space Ext"(M, 45(9)) is absolutely inseparable. Proof. Let (11) be a free resolution of the module M. Then the module L = Coker p3, satisfies the conditions of Theorem 3. Therefore, the lowest row of the diagram (12) is exact, and this implies equation (14). If the module M satisfies the conditions of Theorem 4, then as we showed in

the proof of that theorem, both the spaces in (13) are absolutely inseparable. There remains only to observe that the right space is Ext‘ (M, d) (9)). [I § 11. Geometrical conditions of M—convexity In this section, we prove a theorem which contains the geometric conditions on the functional space «15, under which the equation Ext'(M, d5): 0, i>k holds for an arbitrary hypoelliptic 9-module M. This result may also be treated as a condition for L-convexity of the space r15, where L is the module connected with M by the sequences (10.10)— (11. 10). It'15 clear from these sequences that for arbitrary i>k, the 150morphism (2) holds.

§ll. Geometric-l conditions of M-convexity

347

1°. k-convexity of a region. Let k be some integer lying between 1 and n. A function ME) defined in some region Q, and belonging to the class C2 in this region, will be called k-convex if it is real and if at every point {252, its Hessian Hessh=

{ 01h

} has no less than k positive

6516:}

eigenvalues. We shall say that the region {2 is k-convex if we can define in it some continuous function h which is k-convex outside some compact KCQ, and such that for arbitrary real c the set Kt: {5e52, h(§)§c} is compact. We shall say that the region 9 is completely k—convex, if it is k-convex and K = Q. It is easy to prove that every convex region is n—convex.

2°. M—cohomologies in k-convex regions Theorem 1. Let M be a hypoelliptic module, and let k, 0§k§n be some integer. I. If the region Q is (n—kyconvex, the spaces Ext‘(M,45(Q)), i>k, are finite dimensional. II. If the region Q is completely (n—kyconvex, then Ext‘(M, @(Q))=0 for i > k.

Let us take note of some necessary facts. If the module M is hypoelliptic, Corollary 2. {59 implies the isomorphism

Ext‘(M, 9‘62»; Ext-'(M, 6(0)),

:20.

Therefore, it is sufficient to prove the theorem only for the space @(D)= 6(0). Let pm ykitLykL..._fliyxiiy‘°aM—>O

(1.11)

be a free resolution of the module M. We consider the module L= Coker pg. It is clear that the mappings ..., 1);“, p; in the sequence (1) form a free resolution of the module L. Hence, it is obvious that for an arbitrary 9-module ¢

Ext‘(l.., ¢)§Ext"*i(M, (P),

i; 1.

(2.11)

The remaining mappings in the sequence (1) form an exact sequence (10.10). We take note of the following transparent geometrical lemma.

Lemma 1. In R”, there exists a countable system of closed parallelepipeds 1t={1r,}, having the following properties: I. For arbitrary 8>0, those 1t,l having a diameter less than 5 form a covering of R".

Vll. Inhomogeneous Systems

348

11. An arbitrary set of k parallelepiped: 1!” of which no one contains any of the others, intersects in a set of codimensian not less than k— 1. (The empty set will be given an arbitrary negative dimension.) Lemma 2. We set {‘=(é,, ..., if...) and €"=(€m+,, ..., in), where m=n—k. Let to be a region in R", such that its section by an arbitrary subspace of the form §"=const is a convex region. Then the space 6(a)) is strongly L-convex. For the proof, we construct an increasing sequence of regions tending to a), in each of which Theorem 3, § 10 is applicable. Let K be an arbitrary compact belonging to a). We choose some compact K’ such that K c c K’cw, and such that the intersection of K’ with every subspace of the form £”=const is convex. We write p=min(p(K, CK’),p(K’, (2(a)).

Making use of Lemma 1, we cover the coordinate subspace R’éu with a system of parallelepipeds (1t,}, the diameter of which does not exceed p/4, having not more than k+1-fold mutual intersections. For every a we choose a convex neighborhood u, of the parallelepiped nu, the diameter of which does not exceed p/2, and such that the regions a, intersect in no more than k+1-fold fashion. Further, suppose that it is a projection ofK’ on Rgu ‘ Ifthe given region a, intersects K, we choose some point Elem/WK and we consider the region 12,, which is the intersection of int K’ with the subspace 5”=§;’. If, however, u,nK=Q, then we write v,=Q’. The regions w,=v,xu, are convex, they belong to a), they have no more than k+ 1-fold mutual intersections, and they form a finite covering of the region roK=u w,, containing K. The regions Rg'ixu, are also convex, they intersect in no more than k + 1—fold fashion, and they form a locally finite covering of the space R". Therefore, on the basis of Theorem 3,§ 10, we conclude that the space flaix) is L-convex, and the subspace JAR”) is dense in 51.0%)Since the subspace EL is dense in é’L(R"), it is dense also in 64t and therefore, the space 6(0),) is strongly L-convex. It follows from the construction of the regions noK that K c Lox c ca). Since the compact K :0) was chosen arbitrarily, we may apply this construction to the sequence of compacts K,, v=1, 2, ..., tending to to, and so constructed that for an arbitrary v we have (0,“c +1. The sequence of regions can is an increasing sequence and it tends to a). Since for arbitrary v, the space 6' (tab) is strongly L—convex, Proposition 2, §7, implies that the space 5(a)) is also strongly L-convex, which is what we were to prove.

I]

3°. Proof of Theorem 1. We shall suppose that the region a is n—kconvex, and ME) is a corresponding function continuous in [2, and n—k-convex in 0\K, where K :9 is some compact. By hypothesis, for an arbitrary real c, the set Kc, consisting of the points of {2 at which

511. Geometrical conditions of M-convexity

349

h(€)§c, is compact. We denote by 9‘ the subregion of 9, in which h({)0 in RV)“ (iii) the functions e, are small enough so that for arbitrary or and B, a the matrix Hess,» h.(.f), where

h,=h—z e‘, l

is positive definite for all 55 U‘, where n=(r]',r]") is a local system of coordinates corresponding to the sphere U’.

350

Vll. Inhomogeneous Systems

Further suppose that A is the largest of the numbers a: such that the intersection Km U“ is not empty. For every 0:, ogag A, we consider the region

93= (659, Mike}Frorn (ii) it follows that the function 11,. is strictly less than h in the neighborhood of the set Kc\i'2,. This implies that c Qg‘. Consequently we have the relation [1”,c for sufficiently small positive e=e(c). We shall show that the regions Gamma“, az=0, ..., A, satisfy the conditions of the Lemma. In fact, the relation (3) clearly holds. We take note oftwo properties ofthe regions that we have constructed: a) For arbitrary or, £2"‘\Q““1 c U“, b) For arbitrary a and t3, the space 6(9‘n U") is strongly L—convex. Property a) follows from (i), since h,,[—h,l_l = —e,. Let us verify property b). We choose a and fl arbitrarily. By construction, the matrices Hess", h and Hess": h, are positive definite within the sphere U”. Therefore, the intersections of the regions 9,,41 U’ and {EnU‘ with an arbitrary manifold of the form r,” = const are convex. Therefore, the intersection of the regions Q‘nU"=Q‘:nQ€HnU’ with such a manifold is also convex. Therefore property b) follows from Lemma 2. From a) we derive the equation 9‘“u(£2‘n U‘)=0‘, where a is an arbitrary number lying between 1 and A. It follows that the regions 0‘“ and 9‘ n U" form a covering of the region Q“. The sequence (19.2), when applied to this covering and the sheaf a! is written as follows: 0—>6(£2‘)—">6(Q“‘)®6(Q‘n U"‘)—'—§J(Q‘“l n U')—>0. The mapping ,1 carries the function ¢ into the pair (¢’, ¢”), where ¢' and 45” are restrictions of ()5 on 9"“ and (I‘m U“, and the mapping a' carries the pair (45, (11) into the difference 45' —¢t’, where ¢’ and 111’ are restrictions of d) and w on 9‘“ n U". Proposition 9, §2 implies that this sequence is exact as a sequence of mappings of topological ?—modules. Therefore, we may write the corresponding algebraically exact sequence for the functor Hom(L, ~):

--—“‘—‘> Ext‘(L,af(Q‘))—E—> Ext‘(L, arr-U) a; Ext‘(L, mm (1‘)) 41+Ext‘(L «arm (1‘))i»

(411) '

in which all the mappings are continuous (see Proposition 8, § 3, Chapter I). It follows from b) that Ext‘(L, £(Q’nU‘))=0 for all 1' >0, a, 3. Therefore, the sequence (4) may be divided into the fragments

0—»Ext‘(L,6(D‘))i>Ext‘(L,6(0“))—>0,

i=2,3,...,

(5.11)

§ll. Geometrical conditions of M-convexity

351

and

0—»6L(tr)i>aL(m-l)emmn U") 611 , . Lima-1n U“)L»Ext‘(L,a(m)i»Exti(L,a(rr-'))—»o.( ) The algebraic exactness of (5) implies that for i> l, the mapping r; is an algebraic isomorphism, and the algebraic exactness of (6) implies that r: is an algebraic isomorphism. By b) the space EL is dense in 6119““ n U“). On the other hand, EL belongs to the image of the map— ping a“ in (6), therefore the image of a‘ is dense in 6162“" n U“), that is, the cokernel of 0* is an absolutely inseparable space. The mapping 6°, being continuous, acts continuously from the space Coker «7* to the space Ker r11. Therefore, the space Ker r: is also absolutely inseparable”. Thus, the properties A) and B) of the mappings r; have been established. We now prove property C). We shall suppose that the space Ext1(L, t! (9‘)) is separable. A separable space cannot have any inseparable subspace. Therefore, B) implies that Ker r§=0. We consider the commutative diagram 0

0

t . l

0

——>

5L(9“)—"-’JL(9‘")

aL(rr)—"-»al(n~-l)eamtrn U‘)L>6L(Q"1 n U“)——>0 —R

[I

w'LWnU‘)

-—* 6L(Q"nU“) -—>0

in which R is a restriction mapping, and i and 1: are mappings acting according to the formulae

1': 05—4045),

1r: Mum-'41.

The left column and the lowest row of this diagram, which consist of identity operators, are exact. The exactness of the second column is obvious. Since Ker r:=0, the algebraic exactness of (6) implies the algebraic exactness of the second row. Since this row is formed of Frechet spaces, it is also topologically exact by Proposition 8, § 1. Thus, we may apply to our diagram Theorem 1, §2, Chapter I, which implies the topological isomorphism

Coker r: ; Coker R .

(7.11)

13 The image of an absoluw inseparable space under a continuous mapping is again absolutely inseparable This can be verified immediately.

VI]. Inhomogeneous Systems

352

Since the space 5,19% U‘) contains EL, which by b) is dense in 6162‘“ n U“), the image of the mapping R is dense in 5119'" n U"). Therefore, the space Coker R is absolutely inseparable Taking account of the isomorphism (7), we conclude that the space Coker r? is also absolutely inseparable, which is what we were to prove. I] We consider the numerical sequence

c1 =0,

ez=c1 +£(Cl), ..., cl“ =c‘+e(ez), ...,

where 5(a) is the function constructed in Lemma 3. We shall show that this function can be chosen so that cA—wo. In fact, the number 5(a) is defined by the condition that the region ‘2”, belongs to (If. The construction of Qg‘ makes it obvious that this region contains a 6-neigh— borhood of the compact Kc, where the quantity 6:6(c)>0 can be assumed to be a non-increasing function of c. The number e(c) can be made to satisfy the condition: e(c) is the largest of the numbers 5 for which the region 9,“ belongs to the region a)“ which is a 6(c)-neighborhood of the compact K,. We shall show that in this case ell—mo. Let us assume the opposite. Then, cA —r c‘I < co, and therefo_re, for arbitrary A, we can find a point clan“, not lying in a)“. Since 9,. is compact, the sequence {:1} has

some limit point 6,, e 9“. 0n the other hand, the union ofthe compacts K“ is equal to ‘2“. Therefore, the point 5,, belongs to a 6(c,)-neighborhood of some compact K01 and therefore, belongs to the region a)“ for all sutficiently large A. We have thus obtained a contradiction with the relationship gem”, It follows that cA—too, which is what we were to prove. Lemma 4. Let (0 and DD 30) be arbitrary bounded regions in R", such that the restriction mapping

(8.11)

Ext'lM, 6 (0)) —> Ext‘(M, J ((0))

is an algebraic epirnorphism. Then the space Ext'lM, ((111)) is separable and finite dimensional. Proofi Let V: { K} be some locally finite convex covering of the region 0, and U: {UV} be a locally finite convex covering of the region a), such that for some 6>0 the covering formed of 6-neighborhoods of the regions U' is inscribed in V. We denote by ZMV) the kernel ofthe cobound— ary operator 6‘: ‘6“ V)—v‘*‘3M(V). We give a similar meaning to the expression ZMU), and we consider the mapping

3?: Zh(V)e"‘5u(U)—>z:,(v),

i>k,

(9.11)

N l. Geometrical conditions of M~convexity

353

which is equal to the sum R+A, where R and A are operators defined by the formulae

.{ZMW

—">Z‘u(U)

' “Wm—a;

{ MV)

—+0

' "-‘tM(U)—£‘—~ZL(U).

and p is a restriction mapping. Since by hypothesis, the mapping (8) is an algebraic epimorphism, Theorem 1, §9, implies that the restriction mapping H'(V, 6“) —v H‘(U, if") is also an algebraic epimorphism. It follows that the mapping 9? has the same property. Let us now consider the restriction operator p: ‘6’M(V)—> l6’,.,(U). We may look on it as a mapping acting from the direct product

Wham-an.) to the direct product H ("(Um n - - - n U“). The mapping ofdirect products differs from zero only on those factors 6“a ~-~n V“) for which the intersection Vvon n V,” no) is not empty (the number of such factors is finite, since to c c v) and it acts on these factors as a restriction operator on a subregion of the form UM n - - - n Um . which together with its 6-neighborhood is bounded and belongs to V," A ~ -~ n V,,,. By Theorem 1, §5, every such restriction operator is compact. This implies the compactness of the operator p and, therefore, of the operator R. Thus, A =Q—R, where 9? is an algebraic epimorphism, and R is a compact mapping. Since all the spaces appearing in (9) are Frechet spaces, we may apply to A the Schwartz Theorem “, which implies that the space Coker A = H‘(ll, 4!") ; Ext‘(M, 6(a)» is separable and finite dimensional.

[I

Corollary 1. Let c be an arbitrary, non-negative number, and let e=s(c) be the constant defined in Lemma 3. Then for arbitrary i>k, the restriction mapping

R‘: Ext'(M, 5(1),“ )—» Ext‘(M, mm)

is a topological isomorphism, and the restriction mapping

Ri—1: 5,,_I(QH‘) ‘M’n—c) has an image dense in {“402}. Proofl We fix an arbitrary i> 0 and we consider the restriction mapping Ext‘ (L, 3(1)”J) —¢ Ext‘ (L, {(03). 14 See Schwartz, [2], Part II, 67.

(10.11)

354

VII. Inhomogenq Systems

It is a composition of the operators r: with a=A, A—l, ..., l, which are algebraically epimorphic in view of Lemma 3. Therefore, (10) is also an algebraic epimorphism. We further choose fl arbitrarily, OgfigA, we set s’=s(c+e(c)) and we consider the sequence of operators

Ext‘(L, 619...“,1)» Ext‘(L, m...» —+ Ext‘(L. «719‘»The first of these coincides with (10) when c is replaced with c+e and, is therefore, algebraically epimcrphic. The second is the composition of operators r; with a: = A, A — 1, . . . , ,8 +1 and, is, therefore, also algebraically epimorphic. Therefore, the composition of these has the same property. Since 12h: CKZHHE, we conclude on the basis of Lemma4 that the space Ext‘ (L, N95» is separable By definition, Ext‘(L, 6(9’» is a factor—space of the f—space fiflflm") with respect to its subspace Pim— l[(1f(!2")]"”"‘. Since the factor-space is separable, the subspace is closed. Therefore, the subspace and the factor-space arc Frechet spaces. By Lemma 3, the mapping r: and the mappings r; with i>1 are algebraic isomorphisms and, therefore, by Proposition 7, §l, are topological isomorphisms. Thus, the mapping (10), which is their composition, is also an isomorphism. The mapping R'“ is isomorphic to (10) in view of (2), and, is, therefore a topological isomorphism. We shall now show that for arbitrary a and 12k, the image of the mapping 1 _1

Pas Jp,(9“)-’6,,(9' ) is dense in 61,1(9'4). If i>k, then the fact that rfi is an epimorphism implies that the image of pi, together with the subspace p.._ ,[a'(§2“1)]"-I is equal to the space imbl). But the subspace 5m“) is. clearly, dense in 662‘”), and, therefore, pi,l[6(£2‘)]”"clm p: is dense in pi_l[r?(§2“1)]“". It follows that Im pf, is dense in 6pi(£2““). If i=k, this assertion follows from Property C) of Lemma 3. Since the operator Ri is the composition of the operators p; with a=A, A— 1, , 1, its image is dense in 3mm). 1]

4“. Completion of the proof of Theorem 1. We now fix an arbitrary i>k and A and we consider the sequence of linear topological spaces

0" 5p...(flc.)—> Minn)?“ —"%1z.i_l manly“! —» 0. (11.11) It is, clearly, algebraically exact. As we noted above, p,_1[d'(9”)]"" is a Frechet space. Therefore, (I 1) is an algebraically exact sequence of Frechet spaces. Thus, it is exact. When )1 runs over the sequence of natural numbers. the terms of (11) run over a decreasing family of linear topological spaces, with respect to

§111 Geometrical conditions of M—convexity

355

restriction mappings. In View of the corollary. the space gammy is dense in mmu). Therefore, by Proposition 11, (51. the sequence (11) permits us to pass to the projective limit while preserving its exactness. Since 61—» 00 for A—> 00, we have 9,, —>0, and therefore, the projective limit of the sequence of spaces 6’62“) coincides with 6(0). Then the limiting sequence may be expressed as follows:

ea 6Pl-l (9)4 [6(9)]"4 —»1iy.-.[:(sz.,)1"—-—»o. 1

Its exactness implies that

li7mP1»1[5(91,)]""Elk--1319) "“-

(12-11)

is an algebraic isomorphism. In a similar way, we pass to the limit in the sequence

0 —> P1, 1 [6’(91‘3" " a 6.412..) -> EXt‘iM, 1319.)) -> 0 (the space pi_1 [6(KZEAH)]"", as we have seen, is dense in Pi- 1[6(Q”)]“"). Taking account of (12), we obtain in the limit the exact sequence

0-tp1_1[6(9)]"" 46,.(9)—>li_ifl 5*“i “(Dam—'0A

The exactness of this sequence implies that

gm Ext‘(M,a$'({2cA )gExt’(M, 6(0)).

(13.11)

is a topological isomorphism. In accordance with the corollary, all the restriction mappings R‘: Ext'iM, 1919“,!» a Ext'iM, 662“»,

which are used in the left side of (13), are isomorphisms. Therefore, this projective limit is isomorphic to an arbitrary one of the spaces EXtiiM- 69(95)- Hence, finally

EX'WM, 5(9o))EExt‘(M, .1112»,

i>k.

(14.11)

In view of Lemma 4, the space Ext‘(M, ((00)) is finite dimensional, and, therefore the space Ext‘(M,6(Q)) is also finite dimensional, which is what we were to prove. Let us now suppose that the region Q is completely n—k-convex. Then K=Q, and, therefore, we may suppose that h>0 in 0. Hence no: a and, accordingly, the left side of (14) is equal to zero. Therefore, the right side is also equal to zero. I]

VIlr Inhomogeneous Systems

356

Remark. We may deduce from the proof of Theorem 2 certain facts about the spaces Ext‘(M,é’(£Z)) in those cases also when M is a nonhypoelliptic module. In fact, we note that the arguments of the theorem for i>k+1 do not depend on Lemma4 and, therefore, are true for an

arbitrary module M. Thus, the algebraic isomorphism (14) with i > k+1 holds under the assumption only that the region Q is n—k-convex. To establish this isomorphism for i=k+l, it is sufficient to know that the spaces Ext"+1(M,é’(w)) with we!) are all separable. Thus. we may formulate

Corollaryz. Let M be an arbitrary finite 9-module, and let 9 be an n—k—convex region. Then for arbitrary i >k+1, the algebraic isomor— phism (14) is valid. If all the spaces Ext"+‘(M, 6(a)» with (DC!) are separable, then this isomorphism holds also for i=k+1. If the region [2 is completely n—k-convex, then Ext‘(M,€(n))=O for all i>k+ 1, and, in the second ease, i>k. § 12. Operators of the form p(D;) in domains of holomorphy Let n be an even number: n=2m. In the space R: we define the structure ofa complex space C'g', writing Cj=éj+if_+j, j=1, ..., m. We consrder the differential operators

6 a o =(—,...,,—)

‘

(if,

a a D =(_,,,.,,_._)

35”

z

«951

35.,

The polynomials in the operators %, j = 1, . . . , m, or in the operators 3%. .

.

1

.

j j = 1, ...,m, with complex coeffiments, form subnngs 9’ and .9” of the ring 9. A finite 9-module M will be said to be holomorphic if M ; Coker p’ where p is some ?’—matrix, that is, a matrix formed from polynomials in D2- In particular, the module corresponding to the CauchyRiemann system of equations in C" is holomorphic. In this section we shall show that for holomorphic 9-modules, the domains of holomorphy have the same roles as convex regions have for arbitrary 9-modules.

1°. Formulation of the theorem Theorem 1. Let [1: C2" be an arbitrary domain of holomorphy. Then for an arbitrary holomorphic 9-module M, the spaces ((9) and 9"(9) are M—convex. If [2 is a Range region, then these spaces are strongly M-convex. The basic property of a domain of holomorphy that is needed in the proof is the following: every domain of holomorphy in C'” is a Stein manifold, and therefore, admits a holomorphic, regular. proper imbed—

912. Operators of the form MD!) in domains of holomorphy

357

ding into the space Ch“ ‘5. That the mapping is proper means that the pre-image of every compact in Cm“ is a compact in Q. The regularity of the mapping amounts to the fact that the Jacobian at every point (e!) is equal to m. For convenience, we set v=2m+L The variables in the space CV will be denoted by 1. =0.“ , 1,). The mapping 0 —> C; described above, will be denoted by l. The image of the point C under this mapping will be denoted by l(§)=(11(§), ...,).v(§)). That the mapping 1. is holomorphic amounts to saying that all the functions 11(0, ..., 1&0 are holomorphic in (2. Let us consider the space 0"", which is a direct product of the spaces C2" and C}. We denote by 1: the projection operator from 0"“ on C2", and by A the mapping from £2 to 0"“ which carries the point; into the point (5, 1(0). The operators A and 1: set up a biholomorphic homeomorphism between the region Q and the manifold Am), defined by the system of equations 1:}.(5), (all. It is easy to see that the manifold Am) is a closed set in 0“". For every region U, belonging to the space C'I or to the space 0“", we denote by (MU) any one of the three spaces X’(U), £(U), 9“(U), where JflU) is the space of holomorphic functions in U, with its topology induced by €(U). Let U be an arbitrary region in 0"“, and let V be the projection on C’g‘ of its intersection with Am). We now consider the

opera!“

A‘: «mam; A)—>¢(C,MC>)E¢(V%

which carries the function 4) into its restriction on A(Q), looked on as a function of 41 Clearly, this operator is defined and continuous if @(U)= JflU), and @(V)=.#(V) or ¢(U)=€(U), and ¢P(V)=J(V). We denote by 2 the ring of polynomials with complex coefficients in the operators DA, 0;. The ring 9 of all differential operators in CM" with complex coefficients is equal to the tensor product 9 ® 2 over the C

field C. We denote by .1! the {JD-module M ® Qgfi/p’a', We consider C

the Cauchy-Riemann system in the variables 1:

6n _

_ flu

6/11 _

_ all,

=0.

(1.12)

We denote by L the fi-module corresponding to this system, and by! the Sit-module 9® L corresponding to the same system, considered as c

being in the space C'"+ V. ls See Narasimhan [I]. In fact, for the proof ofTheorem 1, it is sufficient to use a most elementary property of domains of holomorphy: such a domain is the limit of an increasing sequence of Wei] regions. However, the use of the theorems already formulated in the text allows us to simplify the proof.

358

VII. Inhomogeneotu Systems

If U is a region in 0"“, then (PAH) is the space of functions holomorphic in A, belonging to (MU), and mum) is a subspace of it, consisting of the solutions of the system p(Dz)u=0 (here, and later, the symbol ® denotes the tensor product over the ring a). We can now formulate the second fundamental theorem. Theorem 2. If M is a holomorphic 9-module, the sequence 4518;,(C"+')A> ¢M(fl)a0

(2.12)

is defined and exact. Here @(9) is one of the spaces 6(9) or 9" (9), and @(C’HV) is correspondingly, either {(C'H') or 9'10""). 2°. Four lemmas Lemma 1. Let U be an arbitrary region in QXC'. If the function fedigw) vanishes for 11:0, there exists a uniquely defined function getPflU), such that f= 1.1 g. The correspondence f—» g is continuous. Proof. For the spaces @(U)=.#(U), £(U), the assertion is self— evident. We shall prove it for the space 9" (U). We assume. to begin with, that the region U is a direct product of the regions VCR and WcC". Every f 69*(U) and ¢69(V) can be put into correspondence with the distribution (f, ¢)569*(W) by the equation

«I. 05):, W)=(f. Will).

V1690”).

This correspondence defines a continuous mapping 9*(U)—>Homc(9(V), 9*(W)),

(3.12)

where Homc(9 (V), 9* (W)) is the space of all continuous linear mappings from 9W) to 9’(W), with the topology of bounded convergence. According to the theorem on the kernel ‘6, the mapping (3) is a topological isomorphism. The subspace 9;(U)Homc(9(V), X’(W)).

(4.12)

It follows from Theorem 1, §5, that the topology of the space #(W) coincides with the topology induced by 9‘(W). Therefore, the topology of the space Hornc(9(V), #(W)) is that induced by Homc(9 (V), 9*(W)). Thus, the fact that the mapping (3) is an isomorphism implies that the mapping (4) is also an isomorphism. 16 See, for example, Schwartz [3].

912. Operators of the form p029 in domins of holomorphy

359

Let us suppose that the function f e9;(v) vanishes on the subspace Al=0. Then for an arbitrary function ¢69(V). the analytic function ()2 (final/(W) also vanishes for l,=0. Since the lemma is true for the subspace #(W), the quotient g‘=}l.f 1(f, 43); also belongs to sflW). The mapping (f, ¢)¢-—>g, is continuous. We have thus constructed a mapping 9(V)3¢ a has?(W), which, as an element of the space Homc(9 (V), #(W)), is continuous in f. Since (4) is an isomorphism, there corresponds to this mapping some function 369}(U). It follows from its construction that it is unique, continuous in f and that 11 g: f. This proves the lemma for regions of the form U = VXW. Now suppose that U is an arbitrary region in 52x0. We construct a locally finite covering 0” of this region, consisting of regions of the form Uh: kWk. We consider the function f69;(U), which vanishes for Al =0, as an element of the space of cochains ”913911). Using the particular case of the lemma proved above, we can represent the function f in the form 1.1 g, where the cochain g belongs to "95:01”, is uniquely defined, and is continuous in f. We conclude from the equation 6°f=0° 21g: 1.1 6° g=0 that 6° g=0. Since the sequence (25.2) is exact with 45:9", it follows that the cochain g belongs to 9;(U) and that as an element of this space it depends continuously on f. I] Again, let U be an arbitrary region in 0"“. For every k=0,1, ..., v, we denote by (by U) the space of differential forms typified by A “at,

Z ¢i., "qt. “1. A

¢n....,ne¢z(Ul~

We consider also the operator I: ¢}(U)—r¢‘}“(U), which consists of * the multiplication by the differential form 1:2 [Al—A,(C)] d1]. 1

It is clear that 1 1:0.

Lemma 2. Let U bea region in 0"“, “tidying the following condition: U is the direct product of the convex region V=A“(U)c C? and Wc C}. Then the sequence

0-* ¢g(U)—'* “Mm—5

-' ¢}“(U)—" 453(11)’;t ¢(V)H0

(5-12)

is exact, where A. is an operator acting according to the formula

A': (pd/1, A Adl,—>A‘¢. (In the course of the proof, we shall show that this operator is, in fact, defined for all three types of spaces denoted by @(U).) Proof In the region 9x Cv we make the following holomorphic substitution of variables: (6.12) (—1, A—rl’=l—A(O.

VII. Inhomogeneous Systems

360

We rewrite the sequence (5) in the new system of coordinates, and by U’ we denote the image of the region U. It is easy to see that the change of variables (6) defines an isomorphic mapping of the space 45(U’) on the space @(U). Since the system of equations (1) does not change its form, the subspace (P,(U’) goes over under this isomorphism into (DIM). Therefore, for arbitrary k, we have the isomorphism @(U’)§IP§,(U). The operator I is transformed into the operator 1’, whose action consists in multiplying by the form l’=ZA} 11).}. Finally, the operator A0 is transformed into an operator which acts according to the formula

A’: ¢(£, 19111; A

A dig,» (MC, 0).

Thus, the sequence (5) is isomorphically transformed into the sequence

0—> ¢3(U')L> ¢}(U’)Lp

,

,

--~—>@t-‘wuLwyvq4wmw.

(7.12)

When @(U’)=X’(U') or 6(U’), the operator A’ is clearly defined and continuous. We shall prove that it is so for the space @(U')=Q*(U’). In the system (1), the variables I. are to be replaced by 1' and we note that this system is strongly hypoelliptic in 1’, with the exponent i=1 (see §5). Accordingly, by Theorem 3 of § 5, every distribution u defined in U’ which satisfies this system, has a restriction on the subspace A’=0 which is a distribution in the region V. In accordance with the remark following Theorem 3, §5, this restriction, as an element of the space 9*(V), depends continuously on u. We have now established that the operator A': 9;.(U)—> 9"(V) is defined and continuous. It follows that the operator A. in (5) is also defined and continuous. To establish the exactness of the sequence (5L it is sufficient to establish the exactness of (7). We denote by Z*(U’), k=0, v— 1. the kernel of the operator 1': @(U')—> @“(U’) and by Z'tU’) the kernel of A’. The space Z°(U’) consists of all the functions f, belonging to ¢,(U'), such that A}f=0, j: l, , v. Therefore, Z°(U’)=0, when ¢P(U’)=1’(U') or {(U’). When @(U')=9*(U’), this follows from the isomorphism (4). We now construct the continuous (single—valued) operators

l"‘: Z‘(U’)—>¢§“(U’). and the operator

k=1,...,v

(8.12)

A' d: 4’00" QM”),

which are inverse to the operators 1’ and A’ respectively. The fact that they exist implies the exactness of the sequence (7) and therefore, of the sequence (5). We begin by constructing the operator A’ ". An arbitrary function ¢e¢(V) can be continued in U’, as a function which is constant in A.

§12. Operators of the form p(D;) in domains of holomorphy

361

The extension i} so obtained, clearly belongs to dtyw’) and depends continuously on 45. The mapping 45—» J dl’A -A dl’ is the desired inverse of A’. We now construct the operators 1’ “. We observe that this task depends only on the region U', and does not depend on (2 or A. Therefore, we may use induction on v. We shall assume that operators similar to l’ ‘1 have been constructed for an arbitrary region 0 in CM“ ‘. We note that when v: 1, this postulate is clearly verified. For an arbitrary form ¢e¢§,(U’), where 0§k[4>,(n)]"~i>[¢,(D)]~--+~~ ~-->[‘Pg(D)]"-fl->¢g(D)—’1>4’(m)—>0,

w=A"(CU

is exact.

Proofi We cover the cube I] with a locally finite covering U, consisting of regions of the form Ufl= 1/,v where K: CZ', and W,c C}, each of which satisfies the conditions of Lemma 3. We denote by Va covering of the region a), consisting of regions A“(U,). We consider the following commutative diagram

i» t

t i

la i

is l

0—»--~~> ‘Mw i» “mm —> ---—> “mm 3» 1«mo—>0

la l

i6 1

0—v —»°¢,(U) iNew) —» ~v°¢gw) $°¢(V)~vo 0a...__, ¢2(D)i>

l

0

g(D)——>-n_, ¢,(D)—£v @(w)—v0

l

0

l

0

l

0

(16.12)

§l2. Operators of the form ”09 in domains of holomorphy

365

in which we have omitted the square brackets [...]“*. By hypothesis, [I is a convex region, and U is a convex covering of this region. Since the space (ND) is Z-convex in view of the results contained in §8, all the columns of this diagram, with the exception of the last, are exact in accordance with Theorem 1. §9. The exactness of the last column follows from Proposition 9, §2.

By the construction, each of the regions U satisfies the conditions of Lemma 3, and therefore, an arbitrary finite intersection U,o n- -n U,“

of these regions also satisfies the conditions of this lemma. Therefore, Lemma 3 implies that every sequence of the form

0—»

—> [ogwnmn n unmi» ——v ,(Umnu- n l1,_)—‘L,¢(V,o,___‘“)—»o, VHON-u “=A“(U,°n---nUu)

is exact. From this, we conclude that in the diagram (16), all the rows are exact, with the exception of the lowest. Applying Theorem 1, §2, Chapter I, we establish the exactness of the lowest row. I] 3°. Completion of the proof of Theorems 1 and 2. We shall show that the module .1! has a free resolution of the form (4.7), in which p, 11,, p2, .. . are .4?”-matrices. By hypothesis M '5 Coker p’, where p is some 9"matrix. We write down a free resolution for the 9"-module Mu =gll;/pl y”: 049””

pix-1

gnu.._,,_,

M

175 gm;

nl—I’ ,

pl

grrsfiMl/fio‘(

17.12 )

In this sequence, 11, 111, p1, . .. , 11Ll are by construction 9"-matrices. The sequence (17) is now multiplied in tensor fashion by the ring 9’ over the field C, and we then perform a tensor multiplication by 2 over the same field. Since

9"®9'®9=Q. C

C

M"® 9’®.‘2=.lt C

C

we obtain the exact sequence: 0—».9"

p'a- . wo-IH..._'i,w2 pi at p’ gal—>0 19’

which is the desired resolution of the module .l. Using this resolution, we construct a diagram (see p. 366) in which we have omitted the brackets ...]"‘ “. Since all the matrices F are holomorphic, and the operators p.12“ contain differentiations with respect to the variable ( only, we have 19 The sequence is exact since the functor of a tensor product over a field'is met.

Vll. inhomogeneous Systems

366

for arbitrary k and j the relation i,‘=Fk pi. It follows that multiplication by the matrix F, carries a vector consisting of solutions of the system p(Dz)u=0, into a vector consisting of solutions of the same system. Therefore, the mappings F,‘ in the bottom row of the diagram (18)

l

-~~—>¢g(D)

l

A"MED

l? l

pa—i

PI

TF1

In

-~-—»¢,(U)

l

—»--~i»¢,(m)

Pa-l

pa-i

---—>d>,(|:|)

0

0

0

0

l

Armor—m 91—:

In

3» gm) ww-flwguj) Amwao

ll

£45.03)

a ¢Aorin)i’ ¢Ae£(D)‘—’

0

0

l" l" Armor—‘0 l I A ‘plozuji‘q’ @5130) ——'0 l0 l (18.12) 0

—+--vi»¢,(l:n

are in fact defined. The equations pit-1:12; p1 imply that the diagram (18) is commutative in all its rows and all its columns except the last. We shall show that it is commutative in the last column also. For arbitrary j =1, ...,m and arbitrary functions f, holomorphic in El with respect to the variables A, we have 0

6

53A*(f)=5€—jf(t,l(0)

= E

a BIL-(E)

353;.

{BL-@191. BC-l iii/tag} Mg,—

(fl)

6:) '

Repeated application of this equation yields an arbitrary equation p} A*(f)=A*(p,-f), i=0, 1,2, ..., and these equations imply that (18) is commutative in the last column. This proves that diagram (18) is commutative as a whole. Lemma 4 implies the exactness of all the rows of this diagram. except the lowest. We shall prove the exactness of the columns. Since the

§12. Operators of the form p(D;) in domains of holomorphy

operators pm!) and %,

367

,%, act on different groups of variables,

Proposition 5, §4, Chapter IV implies that Ton-(.11. y)=o for all i; 1. Since the region D is convex, Theorem 3, §9, implies that the space (15,03) is .lt-convex. It follows that all the columns of (18) are exact except the rightmost. The rightmost column is exact in the term Qua») and is algebraically exact in the term [¢(w)]‘. Applying Theorem 1, §2, Chapter I, we arrive at the conclusion that the bottom row and the righthand column are exact. We now prove the first assertion of Theorem 1. Let [I run over all

increasing sequences of open cubes tending to Cm“. Since the mapping A is proper, the region w=A“(|:l) runs over some increasing sequence of regions with compact closures tending to Q. The exactness of the right column of diagram (18) implies that for an arbitrary region a) in this sequence, the space ¢(w) is M—convext In particular, the space 6(a)) is M-convex. The exactness of the bottom row of (18) in the last term implies that every function ue£M(m) can be written in the form A' (14*), where u*eJ¢,8,(El). Since the region El is convex, the function u"

can be approximated by functions e‘eE1,8 g in the topology of the space JMWUZI). Since the operator A’ is continuous, we may use functions ofthe form 11* (2‘) to approximate the function u in the topology of ("(w). Since all functions of the form A‘(e") belong to film), we have proved that the space 6M6?) is dense in dual), and this fact. together with the Mconvexity of all the spaces 6(a)) implies that the space 6(0) is also M-convex (see Proposition 2, §7). To establish the M—oonvexity of 9*(9), we need the following assertion.

Corollary 1. Let K be some compact belonging to to. Then for an arbitrary integer q, every distribution uegmw) can be written in the form 1 of a sum uo+v, where 1406936)), and vewfl‘, while ||v||§§?, Proof. Let K c I] be a convex compact containing the set A(tc). The exactness of the bottom row of the diagram (21) in the last term implies that the function u can be represented in the form A‘(u*), where u‘egje 1,03). Applying Corollary 2, §4, to the distribution 14*, we write it as a sum u3+v‘, where u3e9j8,(C"“) and u‘eMfl', and

||v*||‘,'{§ %. We have u=uo+v, where uo=A*(u3)59;(Q) and v=

A“(u*)e[6,fl]’. Since the mapping A‘: [QT—v [6,1]“ is continuous, and the norm of the function 1:“ is arbitrarily small, the norm of v can also be made arbitrarily small. [I

368

VII. Inhomomneous Systems

We shall prove the M—convexity of the space 9‘({2). Since every holomorphic Q-module has a free resolution of the form (4.7), in which are .9”-rnatrioes, it is sufficient to prove that there exists all the p, p‘, a continuous operator (19-12) 9;.(9)—' [g‘ffllT/g; (9),

which is inverse to p. Here, 17 is an arbitrary ?”-matrix and p, is a W’matrix such that the sequence (2.7) is exact. Let B be an arbitrary bounded set in [9(9)]‘. We choose a region wcfl, for which the right column of (18) is exact, and which is sufficiently large so that B belongs to the space [9 (to)? and is bounded in it. The exactness of the right column of (18) implies that we can find a bounded set 3’ in [9(a)]‘ such that for an arbitrary function we9;(m), belonging to the polar of B’, we can find a function u’e[9*(¢u)]', belonging to the polar of B, and such that pu'=w. We shall now suppose that we9;(fl). Then, relying on the fact that the space 9"(w’) is M-convex for some sequence of regions u)’, tending to {2, and relying on Corollary 1, we may apply the arguments of Lemma 1, §8, to prove that the function u’ can be approximated by functions ue[9*(fl)]’, satisfying the same system pu=w. In particular. we may choose a function ue[9‘(9)]‘, satisfying this system. and not exceeding the value 2, on the set B. The correspondence w—v it defines the desired continuous operator (19). Thus the first assertion of Theorem 1 is proved. We prove the second assertion. Let 26C" and IE C2v be variables dual to Z and 11. Suppose, further, that {Nfic C; d“) is a set of algebraic varieties and normal Noetherian operators associated with the module M. As we know (see 4", §4), the set of algebraic varieties and normal Noetherian operators associated with the module L consists of the manifold NL, defined by the system of equations 1’: —i I", where 1’: (ll, ..., IV), I”=(lv+l, "”120, and l=(l’, I"), and the operator 1151. By Proposition 5, §4, Chapter IV. the variety NR‘;>0. We shall establish the sufficiency. Let the dimension of the module M at the point 2 be n—h. We are to show that h>k. We begin by assuming the opposite: suppose that hgk. We choose a variety N), from the set associated with M of dimension n— h and containing the point 2. In C'I we choose a system of coordinates such that the variety N0 is normally placed. Then, by definition, N0 belongs to the variety of the roots of the system of equations

ql(zl,w)=~~=q,,(z,,,w)=0,

w=(zh+1v""zn)l

(1-13)

where q,(z,,w) is a polynomial having unity as the coefficient of the highest power of 2,. We shall suppose that aq/az, does not vanish identically on No. We denote by N: the intersection of N0 with the variety of the roots of the polynomials q/azi, i =1, ...,h. We assume that the point 2 does not belong to N.. Let (9:17)x be a ring of functions holomorphic in the neighborhood of z. We prove that for arbitrary j =2, ..., h, we have

qj0n(q10+"'+qj71 0)=qj(1110+"' +914 02)-

(2-13)

Since by hypothesis, the derivatives aq/az, are different from zero at z, the change of coordinates in the neighborhood of 2 defined by the formula z,—>z}=qj(zj,w), j=1,...,h; w—rw,

is regular. It follows that the ideal ql0+~~~+qj_l(9 in the ring (9 is the set of functions vanishing for z;=~~~=z}_,=0i Therefore, if a function of the form 2i 45, where 4160, belongs to this ideal, 4) belongs to it also. This implies equation (2). We consider the 0-modules Yi=0/q,-0,j=l,...,h. The cohomo-

logical dimension of each of these is equal to one. Therefore, we derive from (2), by virtue of Proposition 3, §3, Chapter I, the equation j-l

Torf (2;, ®z)=0. 1

Using this equation and the remark of §9, we conclude by induction on j that the cohomological dimension of the module i

® 3’; 1 does not exceed j. In particular, the cohomological dimension of the module In

$=®Z=0/(q10+'"+qhw) 1

513. Concerning the modules Ext' (M, .9)

373

does not exceed h. Suppose that 0-—>(D‘"—>0‘h‘I—>->~—t0"—>0'°—>.?—§0

(3.13)

is a free resolution of this 0-module of length h. We consider the commutative diagram

0——v0

I I o

_,0

I I I —. 0..., —P. am:

0—» Hom(M,Y)—>2" _P. 2"

I

A»

I

01:14,

I

(4.13)

m-IAMAHfl-I

I

I I

I I

0i»-m#,grm Aghbl'h

0

0.

The exactness of (3) implies the exactness of all the columns in the diagram beginning with the second. As for the rows, since 0 is a flat 9-module, we may write the chain of isomorphisms Ext'(M,9)®0=fll/p,_,?“"®0§l9m/p,‘l0“"=Ext‘(M,0). Comparing the left and right sides, we obtain

Ext‘(M,9) ® 0 ; Ext‘(M, 0).

(5.13)

Since by hypothesis 2 does not belong to N (Ext‘(M, 9)) for i=0, ..., k, the left side is equal to zero for i=0, , kt Hence, we conclude that for these values of i, the right side is equal to zero, whence we derive the exactness of the sequence

own—ho—u‘ "How—M MAM".

(6.13)

Since, by hypothesis. hgk, we conclude that all the rows of (4) except the uppermost are exact. The uppermost row is exact in View of the relation (19.3), Chapter 1. Applying Theorem 1, §2, Chapter I, we arrive

374

VIII. Overdetermined Systems

at the conclusion that the leftmost column is exact, that is

Hom(M,2’)=0.

(7.13)

We shall now show that this equation, in fact, does not hold. which leads to a contradiction with the postulate that hgk. Let do be a normal Noetherian operator belonging to the set associated with the submodule p’g‘cg‘, corresponding to the previously fixed variety No. We shall assume that it is constructed by the method given in §3, Chapter IV. Then its rows are obtained from the coefficients of the corresponding p’-operator by multiplication with some polynomial. As we know, the null coefficient of a p’-operator is a functional of zero order. Therefore, the corresponding row 6 of the matrix d0 is a differential operator of order zero, and therefore, the transposed column 6' belongs to .?‘. In accordance with the property of Noetherian operators, we have do p’=0 on No, whence p6'|N°=0. By hypothesis, the variety of roots of the system (1) is regular in the neighbort of 2. Since it has dimension n—h and contains the n—h-dimensional irreducible variety No, it coincides with N0 in the neighborhood of 2. Thus, the vector 128’

is equal to zero on the variety (I) in the neighborhood of z, whence p (3'q (9+---+q,, 0. Accordingly, the vector 6’ belongs to the kernel of the mapping p: y‘ey', but is not equal to zero as an element of .9”, since the zero coefficient of a p’-operator does not vanish in N (M). Thus, 6’ is a non-zero element of the module Hom(M, 2’). which contradicts (7),

and therefore, h > k. To complete the proof of sufficiency, there remains to consider the case when zeN*. Since the variety Nt belongs to the irreducible variety No, but does not coincide with it, it is nowhere dense in No. Therefore, an arbitrary neighborhood of the point z intersects N°\N,‘. On the other hand, by hypothesis, the point 2 does not belong to the varieties N(Ext‘(M, m), i=0, ..., k, and, accordingly, a sufficiently small neigh— borhood U does not intersect them. Therefore, we can apply to an arbitrary point {eUr\(No\N*) the arguments set forth above, which lead to the contradiction with the assumption hgk. This completes the sufficiency proof. We shall now establish the necessity, that is, we shall show that

dim, M 0 (ifdimo M g 0, relation (8) has already been proved). Suppose that

(10.13)

p=von~~nvl

is a reduced primary decomposition of the submodule p = p’ ?‘c?‘. We denote by p° the intersection of all the zero dimensional modules p), and by p' the intersection of the remaining modules p1. We set M’ =?‘/p’ and we consider the exact sequence 0—»p’/p—i>M—tM'4v0, in which i is an imbedding. This gives rise to the exact sequence for the functor Ext

—> Exti‘1(p'/p, 0) —> Ext‘(M', 0) —»

,

_, Ext‘(M, 0) _. Ext‘(p'/p, a) _»

(11.13)

The self-evident inclusion r(p’/p):t(p°) implies that the module p’lp is zero~dimensional‘. Relation (8) for this module and the fact that (11) is exact implies the isomorphism Ext‘(M’, 0); Ext‘ (M, 0), 1.5. n — 1. Thus to verify relation (8) for the module M, it is sufficient to verify it for the module M'. Our construction implies that in the reduced primary decomposition of the submodule p’c?‘ there are no zero-dimensional components and, therefore, without loss of generality, we may suppose that in the initial decomposition (10), there are no zero—dimensional components. In C’I we choose a system of coordinates, such that all the varieties

N (pl) will be normally plawd. Since the dimension of each of them is greater than zero, their intersection with an arbitrary subspace of the form zn={ is a variety of dimension one less. Therefore, it follows from

the proof of Theorem 4, §9, that

Tor,(M,L‘)EO,

1:1,

where

L‘=9/(zn—Ot?.

(12.13)

Let 9’ be a ring of polynomials and (9’ be a ring of functions in the variables 21, , z._1 holomorphic at zero, with the structure of the 9’-modules. We consider the 9'~rnodule ML=M®L¢ Lemma 2, §1, Chapter IV, implies that N(M‘) is equal to the intersection of N(M) with the manifold z_={. It follows that dim0 M0 =dimo M — 1. Therefore, by the induction hypothesis

Ext'(M0, 0’)=0, 1 Here we shall suppou that p‘llrp.

iék.

(13.13)

513. Concerning the modules Ext‘ (M, 9)

377

Here and later for brevity we shall write Ext’ (M‘, 0’) instead of Ext;.(M(, (9'). From the exactness of (4.7) and from( 12) we infer the exactness of the sequence , ,

#7.?"LW'AMEAO,

where p;, 171,5.

are restrictions of the matrices p, P»

on the subspace

z_=C. Thus, we have obtained a free resolution for the W-module Mg.

Therefore. the relation (13) implies the exactness of the sequence

0"*--——»'"“'v° (an—W om“,

i=0, ...,k (po=p, p_l=0). (14.13)

We have now to infer the exactness of the sequence 0"” L0“LOJ“",

i=0, ...,k.

(15.13)

Let F be an arbitrary element of the kernel of the mapping )1) in (15). The exactness of (14) implies that Fl,"_o=p,-_l g, where 350"“. We look on the function g as a function of all of the arguments z, and as a constant with respect to 2,. The difference F —p,_l g vanishes on z,,=0, and therefore can be written in the form 2,, I", where F'e‘g“, that is, I’ep,._l 0"" +2, (9“. The equation p, 2,, F'=0 implies that p, F’=0, and therefore the function F’ can be written in the form p‘_l g’+ 2,, F”, where geG‘H. Few". Therefore, 1’6l 0"" + z} (9“ and so on. We have thus

proved that Pep“, 0“" + z: 0" for arbitrary k= 1, 2, ... It follows that the phi-operator 9 vanishes on F, and therefore Fep._l 41"".

I]

2°. Corollaries Corollary 1. In order that dim M < n — k, where 0g k g n, it is necessary and suflicient that Ext‘(M,?)=0,

i=0,...,k‘

Proof The finite 9-module L is equal to zero if and only if N(L): Q. 0n the other hand, for the dimension of an algebraic variety N to be less than n—k, it is necessary and sufficient that at every point z, din-ix N < n —k. Combining these remarks with Theorem 1, we obtain the desired results.

I]

Corollary 2. For an arbitrary finite 9-module M

N(M): U N(Ext'(M, 9)). i— 0

For the proof it is sufficient to set k=n in the Theorem.

3“. The dimension of the modules Ext’(M, 9) Theorem 2. For an arbitrary finite 9-module M dim Ext‘(M,.?)§n—i,

i=0,l,2,....

I]

VIII. Overdetermined Systems

37B

Praafl We return to the notation used in the proof of Theorem 1. We shall represent the module M in the form f/p, and we write the reduced primary decomposition (10) of the submodules pc?’ and we consider the module p’, which is equal to the intersection of all the non-

zero-dimensional components of the decomposition (10). Setting M'= W/p’, we replace (9 by 9 in the sequence (1 l) and we obtain:

—’ Exl"‘(P'/P. 9) —* EXt‘ (M9 9’)-r Ext‘(M, 9) . #Ext‘(P'/P,3’)H

(16.13)

Since the module p’/p is zero-dimensional, Corollaryl implies that Ext'(p’/p, .9)=0 for all ign— 1, Then (16) implies the isomorphisms

Ext‘(M’,9);Ext"(M,9),

ign—l,

and the exact sequence 0—» Ext"(M’, 9) —> Ext" (M, 9) —v Ext"(p’/p, 9) —r 0. Since this sequence is exact, we have N (Ext" (M, 9))=N(Ext”(M’, 9)) u N (Ext" (137;), ?)). By Corollariesl and 2, the varieties N (Ext”(p'/p, 9)) and N(p’/p) coincide. Since the module p’/p is zero—dimensional, it follows that N (Ext"(p'p, 5’» is a zero-dimensional variety. All this implies that if the theorem is proved for the module M’. it is proved for the module M. We may therefore assume that in the decomposition (10) there are no zero~dimensional components. The proof of the theorem will now be carried out by induction on n. Let 0’ be the ring of functions in the variables 2,, ...,z,,_ 1 that are holomorphic in the neighborhood of a point z’eC"“, and let 0 be the ring of functions in the variables 2,, ...,z,,, holomorphic in the neighborhood of the point (z’, {)6 C". These rings will be looked on as modules over the rings 9" and 9, respectively. Suppose that 2’ does not belong to the variety associated with Ext‘(M;, 9’). Then Ext‘(M;, 0')=0. Since in (10) there are no zero-dimensional components, we infer the exactness of the sequence (15) (see the proof of Theorem 1), which is equivalent to the equation Ext’(M, (9)=0. This equation means that the point (z’. 4:) does not belong to the variety associated with Ext‘(M, 9). Thus, we have established the implication whence

z’§N(Ext‘(M‘. 9)) = (z’, C)éN(Ext‘(M, 9)), N(Ext‘(M‘, 9)): N(Ext‘(M, 9))0 (z,,=c).

(17.13)

§11 Concerning the modules Ext' (M. 9)

379

As follows from the induction hypothesis, for arbitrary CEC‘ the dimension of the variety N(Ext‘(M‘, W» does not exceed n— 1—i. The inclusion (17) implies that the dimension of the variety N (Ext‘(M 9)) does not exceed n—i I] The result which we shall now establish is a sharpening olCorollary 2

Corollary3. For arbitrary k 0 Hom(Hom(M, 9‘). 9) into its “second associated" module. This mapping is constructed as follows: to an element f6M there corresponds the mapping Horn (M, 9) #9, which carries the homomorphism ¢eHom(M,.?) into its value ¢(f) at this element. We write E0(M)=Kerju,

E1(M)=Cokerju

and, further, Ei(M)=Ext"‘(Hom(M,9),9),

i=2,3,....

We establish rules for the calculation of these modules. The module E0(M) admits the following simple description: it is equal to the 7:dimensional primary component ofthe null submodule ofthe module M 2. The proof of this assertion is left to the reader. Proposition 1. Suppose that M gCoker p’. Then

E,(M)gExt‘“(Cokerp,9),

i=0, 1,2,

(19.13)

Proofi Throughout this subsection we will use the following abbreviations:

H(M)=Hom(M, 9),

E'(M)=Ext'(M, 9),

:20.

Suppose that

- isflirflli»?—'>?~>Cokerp60 is a free resolution ofthe module Coker p. Since H(M) ; Ker (p: 9‘ —> 9'), the mappings q3, ‘Iza ql of this sequence form a free resolution of H(M). If we apply to these resolutions the functor Horn(-, 9), we obtain two sequences

»ei.wey.i,.... 0”"‘C°"°'P)e9‘i 9-2A.gu is 0""(H(M))A»

(20.13)

which coincide beginning with the term 9'1 (qf is the mapping associated with q’l). From these we derive the isomorphisms E‘“(Coker p); E‘(H(MJ), :21. And this proves (19) for all ig2. 2 We have met this module in § 8 as adescription ofthe virtually hypoelliptic operators.

613. Concerning the modules Ext' (M, 9)

381

We now consider the diagram

0

0

I —>E,(M)

(I)

0—» o

I

ifmixture: p)—>0

——»H(H(M))A.Kerq;

I

I»

, I-

0—sE‘(Cokerp)——>M

I

#Imq’.

I

0

—> Ea(M)

I

—»0 —>0

I —> 0

I

o

0.

Its central square is commutative, that is, qu=iq’,, where i is an imbedding. Therefore, there is defined a mapping 1}} associated to qf. Thus, the diagram is commutative and the first and third rows, as well as the second and third columns are exact. Applying Theorem 1, §2,

Chapterl twice—first to the lower left corner and then to the upper right comer, we establish the isomorphism Eo(M);E‘(Coker p) and we prove that if is also an isomorphism. [I The criterion that we spoke about earlier is formulated as follows. Theorem 3. In order that the module M be included in an exact sequence of the form 09M

n

17:,

1:

gr;

I;

u—i

migsku,

(21.13)

it is necessary and sufieient that E‘(M)=0,

i=0,l, ...,k—l.

(22.13)

By Proposition 1, the necessary and sufficient condition can be formulated as: Ext‘(Coker p,9)=0, i=1, , k, Proof. Suflieiency. Suppose that the equations EI(M)=0, i =0, ..., k—l are satisfied. We shall suppose to begin with that k=l. Then by the definition of E0(M) the mapping j“: M—>H(H(M)) is a monomorphism. The mapping q,“ in (20) is also a monomorphism (the exactness of (20) in the first two terms follows from the property of the functor

382

VIII. Overdetermined System

Ham). Therefore, the composition of these mappings qM: M a?” is also a monomorphism. Now suppose that k>1, Then the sequence (20) is exact in all terms beginning with .9" and H(H(M))gM, which implies the exactness of (21). Necessity; The case k=1, 2. We shall suppose to begin with that in the sequences (21) k=1. We write 11:9", L=Coker ql and we consider the exact sequence 0—»MLJI—‘>L—>0, in which a is a canonical mapping. Applying the functor Hom(-,9), we obtain the exact sequence 0

H(L)

H

H(M)

E (L) ‘ 0

,

(23.13)

—>E'(L)—>0—»E'(M)—)E‘+1(L)—>0—> (where for an arbitrary mapping ,8, we write B*=Hom(fi,.‘9)). Applying the functor Hom(4,9) again to the three terms of this sequence, we obtain the commutative diagram

0—>H(H(M))—'£>IIJLH(H(L))

in. O——->

M

H Len-l»

in L

(24.13) —»0,

Its commutativity implies that q1 = qf‘ju. Since 11‘ is a monomorphism, jM is also a monomorphism, that is, E0(M)=0, which is what we were to prove. We shall now assume that k=2. Then by what we have already proved, jL is a monomorphism. Hence, since the first row of (24) is semi-exact, we have Im qf*2. By substituting L in (27) in place of M, we obtain in particular dim E1 (L) én—k, whence by Theorem 1, E"(E1 (L))=0, i=1, . .. , k— 1. Taking account of these relations in (26), we obtain the isomorphisms Ei(M)=Ei"(H(M))§E“‘(Q),

i=2,...,k—1.

(28.13)

Since by the induction hypothesis E,(L)=0, i=0,,..,k—2, we have H (H(L));L and, therefore, the mapping ll** in (25) coincides with t! and is thus an epimorphism. Hence, E‘(Q)=0. On the other hand, if k>3, the exactness of (25) implies that E‘+‘(Q):E‘(H(L))=O for i: 1, ...,k—3. This proves that the right sides of (28) are equal to zero. This brings us to (22). l] Corollary 4. In order that the module M be imbeddable in an exact sequence of the form (21) of length k, it is suflicient that it be representable in the form ?’/p’.¢, where the matrix 1: satisfies the condition

dirn(z: rangp(z)0 are underdetermined, and the operator do is determined, and when n>l

is overdetermined. We shall now prove that the determined operators are characterized by the fact that the corresponding system (1.4) has no solution with a compact support. We shall further prove that for the overdetermined operators every system (1.4), defined in the neighborhood of the boundary of a convex region, has a unique extension throughout the whole region. Proposition 1. In order that the system (1.4) have no non-zero solution which is a distribution with compact support, it is necessary and suflicient that the operator p be determined. Proof. By (193), Chapter I, the module Hom(M,?) is isomorphic to the kernel .2, of the mapping p: fay. We shall assume that the operator p is underdetermined, that is, this kernel is different from zero. Let F be a non-zero element of the module 9,, and let 6 be the deltafunction in R". The distribution F(i D)6 is different from zero, has a compact support, and is a solution of the system (1.4). This proves the necessity. Let us prove the sufficiency. Since Hom(M,?)gyp, and the 9'module J‘(R") is flat (see Corollary 4, §8), the sequence

0 a Hom(M,?) 8) 6*(R") -> [6*(R")]' —'> [3’(R")]‘ is exact. In this sequence, the kernel of the operator p is the space of all solutions ofthe system (1.4)having compact supports. IfHorn (M,9) =0, this space consists of the single function which is identically zero. [I 2°. Characterization of the irremovable singularities of solutions. Let Q be a region in R", and K :20 be a compact. We shall say that the

388

VIII. Overdetermined Systems

distribution u is a solution of the system (1.4) in {2, having a singularity on K, if usQfAfAK). In the space 9;(0\K) we consider the subspace M ), consisting of limits of sequences of the form {1459149)}, which are stable outside an arbitrary neighborhood of K. We shall say that the singularity of the function u59§(Q\K) is removable, if u belongs to 9,310). The factor-space 9fi(Q\K)/9,f,(fl) characterizes the set of irremovable singularities of the solution of (1.4) in 9 that belong to K. Let u be an infinitely differentiable solution of (1.4) in f2\K, i.e., 1466M (Q\K). We shall assume that the singularity of u on the compact K is removable, i.e., ue m9). Then by Example 1, §15, there exists a sequence of functions u,eé’M(Q) tending to u, and stable outside any neighborhood of K, that is, the function it belongs to the subspace film), constructed in the same way as the subspace m). Thus, the factor-space /\

:M(0\K)/6M (Q) characterizes the irremovable singularities, belonging to K, of infinitely differentiable solutions of (1.4). Both the factor-spaces that we have under consideration will be denoted in the same way: /\

‘Pu(fl\K)/¢M(9)~

(1.14)

Our immediate aim is to describe them. We first obtain some conditions for an element of (tum) to belong to (Dual). /\

Proposition 2. If M is a determined module, we have ¢M(fl)=¢u(fl). If K is a finite set, the inclusion relation

mmteuaneemm. holds, where the left and the right sides are looked on as subspaces in 9:,(Q\K). Proof. Let (14,645" ((2)) be a sequence stable outside any neighborhood of K. Without loss of generality, we shall suppose that "151125"outside some compact 1’ Cf). Then for arbitrary at the function u, — um+1 has a compact support and satisfies the system (1.4). If M is a definite module, by Proposition 1, we have uaau, +1 and, therefore, u=lim 14,; medium). Let us prove the second assertion. Clearly, we may limit ourselves to the case when K =0 where 0 is the origin of coordinates. Let u belong to the intersection mm [9*(Q)]‘. Then pu is a distribution whose support is the origin of coordinates. Accordingly, pu=F(iD)6, where 1769‘. On the other hand, pu=p(u—ul)ep[l‘(R')]’. Therefore, the

§14. The extension of solutions of homogeneous systems

389

polynomial F belongs to the space p[6*(R")]‘ and because of Proposi— tion 4, §1, Chapter II, it has the form pG, where 659‘. it follows that the function u—G(iD)6 satisfies the system (1.4) in Q and, therefore, belongs to the space 93,62). Hence uegfim), if 9,7,(0) is considered as a subspace in 9;(Q\0). [I

Let us consider some spaces of entire functions. Let 1’ be a convex compact in R". We denote by 6”” the space of entire functions in C", characterized by the fact that each of them for arbitrary e>0 satisfies

the inequality

545(2):: C(|z|+1)" exp(€ wow—y)

(2J4)

for some b and C, depending on e. The subspace in 6’”, consisting of functions 4), which satisfy this inequality for arbitrary (>0 and b> —ao, will be denoted by S’”. Let 4 be a 9-matrix. We denote by 4"” {q} and S” * (q) the spaces of q-functions holomorphic in C", which satisfy the inequality (2) for arbitrary e>0 and b, depending on e, or respectively, for arbitrary s> 0 and b > — co. Let L be a finite 9-module. Using the construction of 4°, §5, Chapter IV, we can set up a natural isomorphism between the spaces 61* (q), for which Coker q’sL. By 6’” {L} we denote the space obtained by identifying all the 6’“ {q}, for which Coker q’zL. The symbol SJ“ {L} has a similar meaning. Theorem 1. Let M be a finite 9-module. Let K be a convex compact, and let 9 be a neighborhood of it. Then there exists an operator /\

B: 9;(9\K)/9*(9)—>J‘+{Ext‘(M,9)},

(3.14)

which establishes an isomorphism between these spaces. The restriction /\

B: JM(Q\K)/6M(§2)—> S“ (Ext‘(M,?)}

(4.14)

of this operator is defined. It is also an isomorphism (Here and later, all the spaces will be endowed with the discrete topology.) Proof Let (4.7) be a free resolution of the module M. By definition Ext1 (M,9)=?n/p?’,

9,1: Ker {p12 9' —» WI}.

(5.14)

Therefore, the sequence

O—v Ext‘(M,9) —+ 97p? #9“

(6.14)

is defined and exact Let K be an arbitrary convex compact in R' with a non-empty interior int 1’. We choose a strongly increasing sequence (1;) of convex compacts tending to 1’. We consider the family of majorants

fl={M.(2)=(IZI+1)"Jn(-y), i=1. 2, ...}.

Vlll. Overdetermined Systems

390

Using this family, we can construct for an arbitrary finite 9-module L the corresponding family of spaces 6”(L}=.3t§,(L}. Proposition 6, §3, implies that J! is a family of majorants of type 1. Therefore, by the basic theorem of Chapter IV, we have that the functor L~ —. 6"{L) is exact. Then, applying this functor to the sequence (6), we obtain an exact sequence of families. Before writing out this sequence, we note that in View of the same theorem, we have the isomorphism a" {9711?} ;[6"]‘/p [at]! where d” =6” {9}. Therefore, the exact sequence that we now have in mind can be written as: 0 a 6* {Ext1 (M,.9)) —» [6"]711 [6"]‘ A» [6"]?

(7.14)

For an arbitrary module L, we denote by 6”" {L} the inductive limit of the family 6"{L). Let us pass to this limit in the sequence (7). By Proposition 10, §l, the sequence of spaces

0—> «V- {Ext‘mm a [61171: [fl-ri» [6’1“ (7214) that we so obtained is exact. Since it is exact, the space a” {Extl (M,?)}, in which we are interested, is isomorphic to the kernel of the mapping 1),. Let us write out this kernel. Let 0={6‘,}.=0, ..., I} be a normal Noe-

therian operator associated with the matrix p. By Proposition 1, §4, Chapter IV, we may write 60:1)“ where do is a component of the operator 3, corresponding to the n-dimensional primary component of the submodule p?’ :19‘. Using Theorem 2, §5, Chapter IV, and again using Proposition 10, §1, we arrive at the conclusion that the mapping

[WT/prfl-r—‘hrx—(p) is an isomorphism. Therefore, the kernel of 111 in (7’) may be identified with the subspace 6’" ‘ {17}, consisting of p-functions for which the component corresponding to the operator 6., is equal to zero. We denote this subspace by 6:11;}. We have thus established the fact that the

”mm

A'"{Ext‘(M,9)}—d+€:f‘{p}

(8.14)

is an isomorphism, Let us now construct the operator 8. Let X be a convex compact such that 1/: :K. We choose a function «159(0), equal to unity in some neighborhood Vof the compact K, and having a support belonging to int 1’. Let u be an arbitrary element of the space 9;(fl\K)i The function p(iD)m u is, clearly, equal to zero in the region V\K, and it may, therefore, be extended from Q\K to 9, if we set it equal to zero on K. We denote by E the operator. Since suppEpaucsupp a, Ep at ue[6'(int f)]‘.

§14, The extension of solutions of homogeneous systems

391

Let E pom be the Fourier transform of E pau. Applying the operator —-6, we obtain the desired mapping _ B: u—b—aEpau. This mapping carries the space 9,3;(Q\K) into the space of p-functions holomorphic in C". We shall show that it sets up the isomorphism (3). First, we shall show that it does not depend on the choice of the com-

pact .1”, nor on the function oz. Let a: and oz’ be two arbitrary functions on 9(9), which are equal to unity in some neighborhood V:>K. Since the function (a—a’) u vanishes on V\K, the operator E is applicable to it. Accordingly, Epuu—Epa'u=Ep(oz—a’)u=pE(m—a')u_ Since 11 E(oz —a:’) uep [£*(Q)]’, we have 6E1; a: u — 6E1: Mu =0, which is what we were to prove. Since the support of the function E p a u belongs to int 1’, the properties of the Fourier transform (see Propsoition 2, §3) imply that the function E p a :4 belongs to the space [6"]‘ and therefore, —6E p at us

6"{17}. We note that

,__,

_—

,___,

p1(2)Epau=p1(iD)Epau=Epipau=0, since p1p=0. It follows that the function Bu= —6Epozu belongs to the subspace 6:,” {p}. Taking account of the isomorphism (8), we may look on Bu as an element of the space 6’" (Ext‘(M,9)}. Since this function does not depend on the compact 1’, it belongs to the intersection of all of the spaces a“ ‘ {Ext1 (M,?)) with f 3 :K. This intersection, clearly, coincides with a!“ {Ext‘(M.9)}. Thus, we have constructed the linear operator

B: 959mm) —» a“ {Ext1(M,9')}. We now determine the kernel of this operator. Let Bu=0, that is,

5E p a 14:0. We choose a strongly decreasing sequence {KV} of convex we find a function ave compacts tending to K 3. For every v=l,2, 9(9), equal to unity in the neighborhood of K, and such that supp av: c. Then supp Ep 0:q cKV and, therefore, Ep mvue[d"‘"]'. The equation BE p av 14: 0E p a 14:0 implies, by Theorem 2, §5, Chapter IV, that E p at, 146 p [6"“‘]‘, whence

Epavu=pvw

vv5[3*(iv)]‘~

It follows that p(a,,u—uv)=0 in the region Q\K. We now consider the distribution uv=E(1——at.,)u+vv. It coincides with u outside of K“ and 3 That is, K,=::K,==m and n K,=K.

392

VIII. Overdetennined Systems

is a solution of the system (1.4) in 0, since u,=u—(a, u—vv), and in. the neighborhood of K, we have u,=v,. Thus, the sequence of distributions 14,59,149) stabilizes to the function u outside any neighborhood of K.

It follows that 1469mm. But this shows that Ker B: MD). Let us now establish the converse inclusion relationship. Suppose that 1169,}(9). Then

Epau=p¢uep[l*(9)]’. whence 0E pmu=0. Let u’ be an arbitrary distribution coinciding with 14 outside some neighbort Vc c0 of the compact K. Choosing the function 2169(9) to be equal to unity on V, we obtain

6Epotu’=6Epazu=0. It follows that an arbitrary distribution in the space 91%?) belongs to the kernel of the operator B. We have thus established the equation Ker B: “1(0). It follows that the mapping (3) is biunique. Let us now construct the inverse mapping. Let F be an arbitrary element of the space 6‘” {Ext‘(M,9')}. This space is the intersection of the spaces 6‘“‘{Ext1(M,?)), v=l,2, Taking account of the isomorphism (8), in which f=K,,, v=1,2, ..., we may think of F as an element of an arbitrary one of the spaces 55"117), v=1,2, Then Theorem 2, (55, Chapter IV, implies that for arbitrary v we may find a function Mean-T such that F=a¢,,. Since by hypothesis the component of the function F, corresponding to the operator 00, is equal to zero, we have p,(z)lp,=0. Let d), be the inverse Fourier transform of the function 111,. By Theorem 2, § 3, we have ¢,e[6" (R")]', and supp (15,: K, and 1110 D) ¢v=0. Taking account of this latter relationship, and of Theorem 1, {58, we may find a distribution u,e[@*(R”)]‘ such that

110' D) My = d),Let us now construct the sequence of distributions u;e[9‘(R')], v: 1,2, ..., such that for an arbitrary v, we have u’v “=14; outside K' and pu’v=¢>,, and u',=u1. Let us agree that the first v components of this sequence have been already constructed. Then we construct the function 11,“. Since for arbitrary v we have a¢,=r and W,e[d’"‘]‘, it follows that 3w,“ — lI/v)=0. Applying Theorem 2, § 5, Chapter IV, to the family

”#:5'“, we find that l/IHr-UIIVEPENV‘J‘, whence ¢.+1—¢v=ppv,

where p, is some distribution with its support in K,. Then p(u,+l—-u’,)= ¢v+1—¢,=ppv, and we see that the distribution vv=uv+l—u,—p, is a solution of the system (1.4) throughout R". Therefore the distribution u’,+1=u,+,—vv=u',+p, is the one that we are seeking. And with this, the desired sequence {u’,} has been constructed.

§14. The extension of solutions of homogeneous systems

393

Since this sequence is stable outside any neighborhood of K, there exists in the space 9;(Q\K) a limiting distribution u=lim 14". We shall show that the mapping F—> u is the inverse of the operator B. Without loss of generality, we may suppose that Kl c0. We choose a function a to be equal to unity in some neighborhood VI of the compact Kv And we consider the function pa: 14,. In the region V1 it is equal to oh, and in Q\K it coincides with pom, since u=ul outside K1. Therefore pawl: Epau+¢r Hence

Bu= —6Epnzu= —6puul+6$1=66,=6¢1=fl which is what we were to prove. We have now shown that the mapping (3) is an isomorphism. We now take up the mapping (4). Let 1’ be an arbitrary convex compact with a non-empty interior. By analogy with the space 6’", we consider the space S", which is the union of the spaces 8’”, v=1, 2, ((1,) is a strongly increasing sequence of convex compacts tending to 1’). In the construction of the isomorphism (3) we made use of the fact that 6,11% 36,, where .1! is a family of majorants of type J. The space S” ‘ cannot be represented in this form, but it is equal to u 1i_m'.;€’4, where the union runs over all sets of families .14 having the form

Jt={M.(z)=R.(z)Jx.(—y), u=1,2, ...},

and {R,} is an arbitrary sequence of functions, satisfying the conditions of Proposition 5, § 3. The remaining arguments that we used above apply with the obvious changes to the mapping (4). [I 3°. Remark and corollaries Remark. The operator 8, constructed in Theorem 1, is independent

of the compacts K and of the region Q in the following sense. Let K be a convex compact, containing K, and let a) be a neighborhood of it, belonging to 9. Then the diagram /\

9fi(w\K)/9h (w) 4'“ 6'"+ {Ext‘ (M, 9)}

re]

1]

(9.14)

1 A a x 91(0\K)/9}(0)—fi r5 * {Ext (M, W} is commutative; here i is the identity imbedding, and 1;? is a restriction mapping. Corollary 1. Let K=K. Then the mapping 1;? in (9) is an isomorphism. Proof. If rc=K, the imbedding i is an isomorphism. And therefore the commutativity of (9) implies that 17.? is an isomorphism. I]

394

VIII. Overdetennined Systems

Corollary 2. Let K be a non-empty convex compact, and Q be a neighborhood of it. In order that the equation 9,7,(Q\K)=9§(Q) should hold, that is, in order that an arbitrary singularity of the solution of (1.4) on K be removable, it is necessary and suflicient that Ext1(M, 9)=0. Proof The sufficiency of the condition Ext‘(M, ?)=0 follows from Theorem 1. Let us prove the necessity of the condition. Without loss of generality we may suppose that the compact K contains the origin of coordinates. Then the ring 9 is a subspace in If“, so that the p—lunctions of the form 6F, where 1799‘, belong to 6'” {p}. Therefore, the Noetherian operator 6 defines a mapping

emf—harm.

(10.14)

This mapping is a monomorphism. If 6)“ =0, where F 59‘, Proposition 4, § 1, Chapter 11, implies that F = p G, 669‘. Applying the isomorphisms (5) and (8), we transform the monomorphism (10) to the form Ext1(M, 5’) —> 6'” {Ext‘(M, .9» . Therefore, Ext‘(M, .9) $ 0 implies that J“ {Ext‘(M, 9)) $ 0. This proves the necessity. D Corollary 3. Let us again suppose that K is a convex compact and Q is a neighborhood of it. In order that every distribution satisfying the system (1.4) in Q\K have a unique extension in 9 which also satisfies (1.4), it is necessary and suflicient that the operator p be overdetermined. Proof Necessity. Corollary2 implies the necessity of the condition Ext‘(M, 9)=0. The uniqueness of the extension requires that the system (1.4) have no solution with a support belonging to K. By Proposition 1, this means that Hom(M,9)=0. Therefore, p is an overdeter— mined operator,

Sufficiency. Let p be overdetermined. By Corollary 2, for every solution ueQMKAK), and for every compact fnzK, there exists a solution uxeflmm, coinciding with u in Q\.’I. For any two compacts of, 2’ 3 K, belonging to Q, the difference “1—“! is a solution of the system (1.4) with a compact support By Proposition 1, we have uxsug, and therefore, u, coincides with u in Q\K and is uniquely defined by u. l] 4°. One-point irremovnble singularities We now limit ourselves to a narrower problem: We shall determine the irremovable singularities, at the origin of coordinates, of those solutions of the system (1.4) for which we know that they can be extended in the neighborhood of the origin of coordinates as distributions.

§l4. The extension of solutions of homogeneous systems

395

Let 0 be the origin of coordinates in R", and let (2 be a neighborhood of it. The solutions of (1.4) in Q\0, which are extendable in the neighborhood of 0 as distributions, form the space 9:,(9\0)n[9*(t2)]‘. The irremovable singularities of such solutions form a space, which is the image of the natural mapping

ommom [more ammo/6th). The kernel of this mapping is equal to the intersection 9mg 0 [934(9)? and, therefore, coincides with 93(9), by Proposition 2. This means that the space of irremovable singularities of the kind that we are now studying can be identified with the factor-space

9M9\0) n [@’(9)]‘/93§(9)-

(11-14)

Theorem 2. The operator B establishes an isomorphism of the space (1 1) and the space Ext‘(M, .9) (looked on as a subspace in 60+ (Ext1(M, 9»). Proof. Let u be an arbitrary distribution belonging to

9h(9\0) fl [9‘ (0)]‘~ Since it is extendable in Q as a distribution, we have p a u = E p a u + p u. Therefore B u = — a p a u = ail}. Since the distribution 11 u is concentrated at the origin of coordinates, it is equal to F(i D) 6, where F53”, and p1 F= 0. Hence p‘ii: F(z) and p,(z)F(z)=0. Then the function B u is equal to 6F and, accordingly, belong to the image of the mapping (10), i. e., B ueExt‘(M, 9). Conversely, let 6F be an arbitrary p—function, belonging to the image of the mapping (10). Then F E?', and p1 F =0. Then Theorem 1, §8, implies that the system of equations pu=F(iD)6 has a solution in R". This solution clearly belongs to the space 9fi(fl\0)n [9‘(Q)]‘. I]

Corollary 4. In order that the inclusion £M(Q\0) n [9*(Q)]’c9.*,(0),

(12.14)

be valid, it is necessary and suflicient that none of the varieties associated with the module Ext1(M, ?) be hypoelliptic (see §5). We emphasize that both sides of (12) are to be looked on as subspaces of 9fi(fl\0), that is, every distribution concentrated at the origin of coordinates is equal to zero as an element of the spaces in (12). Thus, the inclusion relation (12) shows that every infinitely dill'erentiable solution of (1.4), defined in 9\0, and extendable in D as a distribution, has an extension in Q as a solution of (1.4) Proof. Sufficiency. Suppose that p?‘=po n - .. n p, is a reduced primary decomposition of the submodule rifle? and that NA, 31. ,1 =

VIIli Overdetennined Systems

396

0, ..., l, are the varieties and the normal Noetherian operators associated with the modules p1, and suppose that 110:9", and that 50:11,. Then pm n p, is a reduced primary decomposition of the submodule py‘ca'n and Mln---nM,, where MA=pA/p9‘ is a reduced primary decomposition of zero in the module 9,,p (see Proposition 1, {51, Chapter IV). Since the variety N1, l>0 is associated with the submodule play“, it is associated also with the submodule M1:9,!” 9". Accordingly, the varieties NA, i=1, ..., 1, form the collection associated with the module flu/p 9‘: Ext‘(M, 9). Let us suppose that none of the varieties M(l>0) are hypoelliptic. By Theoreml the operator B carries the space 6M(Q\0)n[9*(9)]‘ into S°+{Ext‘(M,9)}, and by Theorem2 carries it into Ext1(M,.9). Therefore, for an arbitrary element u belonging to this space, Bu=6F, where F 6.9“, and the p-function 61“ decreases faster than an arbitrary power of |z| in an arbitrary band of the form |y|§C. This means that every component 61F of p-functions decreases faster than an arbitrary power of |z| in the intersection of N, and this band. Since by hypothesis the varieties N,1 are not hypoelliptic, the intersection of each with some such band is unbounded, and therefore, the condition that was formu-

lated above concerning the decrease of the functions is meaningful. Let us choose an arbitrary A>0 and consider the function

M(r)=inf{|0i(Z+C.D)F(Z+€)I, 2+EeNA. IZI=r. |€|§U~ We shall suppose that 01F$0. Since the function am is a polynomial, we have m(r)~cr‘ as r—roo for some c and '1‘. On the other hand, what we said above implies that the function m(r) decreases at infinity faster than an arbitrary power of r. It follows that m(r)EO. Therefore BAFEO on N‘ in the neighborhood of some point ZEN; and, so 611720 on NA, since this variety is irreducible. Thus 6F =0, whence by Theorem 2, we have ueflfim). This proves the sufficiency of our condition. We shall now prove the necessity. We suppose that one of the varieties NA, for example, N,, is hypoelliptic. We construct a function belonging to the left side of (12) and not belonging to the right side. By a theorem due to Lechs we can find a polynomial hem which vanishes on N., and such that the variety of its roots is also hypoelliptic. By Theorem 1, §8, the equation h(i D) 12:15 has a solution in R". Corollary 3 of the same section implies that the distribution v is infinitely dif— ferentiable away from the origin of coordinates. Suppose, further, that F is an element of .4", belonging to all the p,1 for 0;). 1, then we can go back to the beginning and replace h by h"). Hence h F= p G, where 669'. Setting u = G(i D) u, we obtain pu=v=hFu=Fhv=F(iD)6.

Thus the distribution u belongs to [9*(R”)]‘, is infinitely differentiable and is the solution of the system (1.4) away from origin of coordinates; therefore, it belongs to the left side of (12). We shall show that it does not belong to the right side. Let us suppose that it does. Then it coincides in K2\0 with some distribution u’eflmg). The difference u—u’ is concentrated at the origin of coordinates and, therefore, has the form G’(i D) 6, G’EQ‘. Hence F6=pu=p(u—u’)=pG’6, that is, F = p G’ e p 9‘, which contradicts the way in which F was chosen. [I

5°. Examples Example 1. Let M =9/p9, where p is a non-zero element of the ring .9. By Proposition 2, § 13, Ext‘(M, mgM. Therefore, the irreducible components of the variety N of the roots of the polynomial p form a collection of varieties associated with Ext‘(M, 9). Therefore, Corollary 4 in this case reads as follows: Any distribution in a neighborhood [2 of the origin of coordinates, which is an infinitely differentiable solution of the system (1.4) in Q\0, will be extendable in [2 as a solution of the same equation, if and only if none of the irreducible components of the variety N is hypoelliptic. Example 2. The operator 6

=— A — l,

p as

A=

"

62

—i

g at}

is not hypoelliptic, and the corresponding variety is irreducible, since the polynomial p is irreducible. Therefore by Corollary 4 every infinitely differentiable solution of the system (1.4) in [)\0, which is a distribution in Q, is continuable in {2 as a solution of this equation. We shall now show that the requirement that the solution be infinitely differentiable cannot be replaced by a requirement that it be differentiable up to a finite order, no matter how large that order. To this end, we construct solutions of (1.4) in R"\0 of an arbitrarily high degree of smoothness, which are distributions in R“, but have irremovable singularities at the origin of coordinates.

39B

VIII. Overdetennined Systems

Let ue@*(R") be a solution of the equation p u=6. Let K be an arbitrary compact belonging to R"\0. Since

an

A —= u

at;

in the neighborhood of K, we know by a property of the Laplace operator 0“ that deg negL u— 2 l

for every compact L: :> K. Hence

62 u
00 faster than an arbitrary power of t. (We may, for instance, choose w to be the Fourier transform of an arbitrary even function ¢69(R‘) with a support in (— 1, 1).) Since the function Illa.) is even, it is equal to $00.2), where 11/0 is some entire function. We consider a holomorphic p-function f(21, Zz)=l/lo(22), (21, z;)eN. It is, clearly, not identically zero, and we shall show that it satisfies the inequality (13). We first establish the inequality

I'13|+|/12|§2(I}-I3 +1).

IIm 1|; C(IIm 1-3l*+|Im lzl’), (15-14)

where C is some sufficiently large number. The first inequality follows from the fact that lllz §|11|3 +1. Since Im A“: |/'L|‘t sin k 4), where 43 =arg A, the second inequality follows from the fact that for arbitrary 45 and sufficiently large C, we have C |sin (M; C(lsin 3¢P+Isin 245E). Let A be any of the roots 1/1; Then on the variety N zl= 1-13, and 22:11, and therefore, (14) and (15) imply the inequality

lf(z1,zz)I=Ix//(l)I_S_p(-%[|A|3+|1I2])exp(C[|Im 113|*+|Im MW) 5 C06 III) eXP(C llm 2F). from which (13) follows for arbitrary k= 1, 2, .. . . Hence f55° + {p}. This proves the non-triviality of S"+ {Ext‘(M,.¢)}. Example 4. We shall find an explicit expression for the operator B when 71:2, and K30 and

_ 1 [Lid-L] _ a

” 2 as

662

a: '

The polynomial p is irreducible, the variety N =N(p) is given by the ‘ equations 21: —izz, and we may assume BEL The operator B may be applied to functions of the form (“1“, can we K, outside differentiable infinitely are these Since . j =0, 1, 2, choose a to be the characteristic function of some compact 1’: :K. We shall suppose that the boundary of 1” is smooth Since the operator p is of first order, we have EpotC‘l'1=C—1’1pat. Hence

35—14: ‘6m—J—l= — [exPflzi’iéD‘l'fzz’igzi] C_}_I%E'd§1d§zL

= - ICXP(€,z2)C"—‘Z—c,dcz.

400

VIII. Overdetermined Systems

Since the distribution aa/aZ is concentrated on 6.9? we may introduce into the integrand on the right side a factor h59(R2), which is equal to unity in the neighborhood of 6.1” and which vanishes in the neighborhood of the origin of coordinates. Applying Stokes's formula, we obtain .

t7

-

Bu= —%I he ”mean-#17211: 4: =% 535;— [hexri(£,zz)§"'“] did; I

=%o;exp(c,zz)c-i-ldz=ji,z£. Thus the operator (1/108 carries the function {4“ into (l/j !) 211. Since this operator is linear and continuous, it carries an arbitrary series of the form 2 (1,644, which converges outside of K, into the series 1;” of i

2172’:

We note that the series Zajt‘l" is the Bore] transform of the series

at

Ell—{2% Therefore, the operator (1/708 represents the transform inverse to the Borel transform. Thus, by Theorem 1, the Bore] transform sets up an

isomorphism of the space a!“ {p} and the space of series of the form ZaJC‘l“, convergent in R2\K. We note that functions of this type are holomorphic in R2\K, and tend to zero at infinity. Conversely, every function which is holomorphic in R2\K and which tends to zero at infinity, may be expanded in a series of this form. Since the operator p is hypoelliptic, we have 9,;(Q\K)=A’M(Q\K) and therefore, the spaces (“WM and S"+ {p} coincide. This implies that the functions belonging to these spaces are characterized by the fact that they satisfy the inequality |f(z)l§ Cs ”ME IN) ltd—Yr, —Y2)= C. “1309 I120 fxz- ‘J’2).

ZEN,

for arbitrary e>0. Hence, it is obvious that the space (“{p} coincides with the space of functions that are entire in C}2 of order no higher than one, and which have indicator diagrams belonging to a compact K", symmetric to K with respect to the axis {1. Thus, we have arrived at the well-known theorem of Polya: The Borel transform sets up an isomorphism of the space of entire functions of order no higher than

514. The extension of solutions of homogeneous systems

401

one, having indicator diagrams belonging to K‘, and the space of functions that are holomorphic in RZ\K, and tend to zero at infinity. 6°. A generalization of a duality theorem of Grothendieck Corollary 5. Let the module Ext‘(M,9) be elliptic. Then the bilinearform (1:, u) = (i, E p at u), defined on 6,(K)>0.

Since the 9-module €(K) is injective and the relation gin/pf; Ext‘(M, 9) is an isomorphism, we obtain the exact sequences

0 —» 651.1(M.9,(K)—“'> «9410—» o. [:(Kuh—“mnm—w;

(VI)

0 —> 6L(K) —t 6,;(K) —> tExl'(M.9)(K) —> 0. Combining these, we find another exact sequence

1:; [6110]" —> 6,.(K) —"'» 64K) —» 0.

(VII)

The function q’v belongs to JAK) and is, therefore, analytic, since the module Coker e’;Ext‘(M,E?) is elliptic. 0n the other hand. for an arbitrary distribution (156 [an the distribution q d) is annihilated by the operator pl since (V) is exact. Therefore the function qq5e[.£‘+]' is annihilated by the matrix p,(z), and we have dq 666'” {p}. By Theorem 1. we have 61175: —Bu for some distribution ue9:(h\K) whence

(11’ 12. ¢)=(iq¢)=;lfiiq6m= —);j(B u), 111=(v,u)=0. But this equation implies that the analytic function 11' v vanishes on K together with all its derivatives. Therefore. the function q’v vanishes in the neighborhood of K, Le, it belongs to the kernel of the mapping q’ in (Vll). Since this sequence is exact, we have vep’l[6(K)]", which is what we were to prove.

514. The extension of solutions of homogeneous systems

403

_ From all this it follows that the form (1), u) is nonsingular, if we cons1der1ts first argument as an element of the factor-space

3,1(K)/P'1[J(K)]”-

(VIII)

Since our form is continuous in the second argument, we have imbedded the factor-space (VIII) in the space of continuous linear functionals on (IV). We shall now verify the fact that this imbedding is an epimorphism. We note that the variety U N is elliptic since it is associated with the

module Ext1(M, 9). Hence it=is easy to conclude that the space 6"“(11) coincides with the space of p-functions holomorphic 1n CI and having zero 6o-components, and satisfying the condition that all the norms

ufux= max sup WI" 121 N. exp(lx(—y))’

(IX)

are finite, where 1’ 3 :K. In at" {p} we introduce the topology defined by the ensemble of these norms. Returning to the proof of Theorem 1, it is easy to detect the fact that the operator B establishes a topological isomorphism between the factor-space (IV) and the space 65* (p), with this topology. Therefore, every continuous functional V on (IV) can be considered to be a continuous functional on 6:? (p). The family of norms (IX) yields a topology of countable type and therefore V is continuous in one of these norms. Further, applying an argument similar to the proof of Theorem 1 of §4, we write the functional V in the form of an integral

(VJ)=Z Ifm, a N;

where the measures [11 have finite integrals of the form (II). Substituting these measures in the right side of (I), we obtain the corresponding function veJP.(K), which in view of (III) coincides with the functional V. We have thus proved that (VIII) is the space of all continuous linear functionals on (IV). It remains to remark that because (V1) and (VII) are exact, the factor-space (VIII) coincides with 55‘111M,9)(K)- [I 7". Special results on the extension of solutions Theorem 3. Let L be a subspace ofR" of dimension n—m, 0 9"—>~~

M7532... r»- grmu

rum 91m:

The condition a) with i=2, ...,m implies that this sequence is exact in the terms M1, 9", ..., 9"". Thus the module MI is included in an exact

sequence of the form (1010) with k=m— 1. Further, applying the arguments used in the proof of Corollary 4 to the matrix p=pm we find that the set of varieties associated with the matrix p”, is equal to the set associated with the module

ExW‘w, 9w;munew plus the variety associated with the submodule QPWFQ‘M', which is equal to C" (or to the empty set, if p“, :0). We now note that the regions (1,: iw, j =1, ..., m+1, are convex, and have no more than m-fold mutual intersections, and form a finite covering of the region 9. Let NA be a variety belonging to the collection associated with the module Mm, that is, associated with the ma-

§14. The extension of solutions of homogeneous systems

405

trices pm. What we have already proved implies that N,| belongs to the collection associated with the module Ext"'“(M, 9), else NA: C". In the first case, condition b) implies the existence of a hypersubspace LA: L, with respect to which N1 is not hyperbolic. It is clear that the projection of an arbitrary region of the form Uhn n Uj... on L1, is equal to the projection of the region Vi] n n VJ". Since the regions V] can be chosen to be arbitrarily close to the faces of the tetrahedron A, their intersections

of the form ln 0 Vi”. can be made arbitrarily close to the vertices of this tetrahedron. We may suppose that the projections of the vertices of the tetrahedron A on L‘, are distinct: if this is not so, we have only to

carry out an arbitrarily small rotation ofA. Then by a suitable choice of the neighborhoods of VjDAj we find that the projections of the interv sections VjIn-u 0V," on Lil are pairwise disjoint, that is, for a given variety N, the condition prescribed in Theorem 4, § 10 is satisfied. The variety NA= C” is not hyperbolic with respect to an arbitrary hyper-subspace in R" and, therefore, the condition of Theorem 4, §10 is satisfied as well. Thus, all the conditions of the theorem are satisfied and,

therefore, the space 6(0) is strictly Ml-convex. Hence, by Proposition 1, § 7 we conclude that

[6‘ (Q)]" n p, [6* (R")]‘ = p1[6*(fl)]'.

(17.14)

The inclusion relation (16) implies that the function 111 B p it belongs to the lefthand side. It therefore belongs to the right side also, that is, Pi fl p a =p1 v, where ve[é‘* (Q)]'. Hence p103 pfi— v)=0. The condition a) with i=1 implies that the sequence

TATAW‘.

is exact. Since the region U=vxw is convex, the 9-module 6*(U) is flat. Therefore, the exactness of this sequence implies the equation J;(U)=p[6*(U)]‘. Since suppUlpfi—v): U, we find that fipfi—v=pw, where we[6*( U)]’. Thus, the function fi—w is the desired extension of the function u. [I Theorem 4. Let £2 be a convex region, let K :0 be a compact, and let 2,, i=1, ..., m, where 0”6§(U)i>"“6’;(v)a

Jaymiwm, z)—.o. (19.14)

is exact. Here “65“(U), v=0, l, ..., m, is the space of cochains of order v on the covering U with coefficients in the space “(”10" n Uh), whose supports belong to the family S. 6 In this lemma we ignore the topology of the linear spaces; the exactness ofa sequence therefore means algebraic enemas. 7 This is the only occasion for the use of the convexity of the region 11. Thus our theorem is proved for arbitrary regions 12 such that all the U, are convex.

§M The extension of solutions of homogeneous systems

407

Let us consider the commutative diagram

0

0

l

l

6*“), D—pt 6"(0, Z)¢r 6*(12, 2)

Wan—M oareal) —>" "l

4

—"~—’p. "-Zastv) (20.14)

4

w—»”"“ ”Wm—M“ p."“:;(U)—»o

.4 an] WNWA» p..'6§(U) ——»o

in which we have omitted brackets of the form [...]"‘. The columns of this diagram, with the exception of the last, are exact, because (19) is exact, The conditions of the theorem imply that the sequence

9:_P,gvi,gt2L,...#v—'I gun-Ayn“.

(21.14)

is exact. An arbitrary region of the form cu=an---nU,-v is convex and therefore, the 9-module 5"(01) is flat, and this implies that the sequence

[J‘(W)]'—"> [6‘(w)]'—“*

-* [5‘(w)]’" A p..[6"(w)]""—>0~ (22.14)

is exact. The exactness of this sequence implies the exactness of all the rows of (20), except the topmost. We shall prove that the righthand column is exact. Let 4: be an arbitrary element of the space p,,I ["“J;(U)]"", such that 6,_1¢=0. In view of the exactness of (19), we have ¢=a,,, 1/1, where

W=X¢h.....1.. ”10A

A QME["5§(U)]'"“~

We shall show that all the distributions dim I" belong to p," [6‘(R")]‘-. Let us fix an arbitrary set of indices jo, ...,i,,. and choose a polygon 1 consisting of segments of the form ni‘n nn‘m, which join the point

408

VIII. Overdetermined Systems

nhn null, and some other point of the form mom A 7th", lying on the boundary of the projection supp III on B. Let us write out the sequence of coefficients of the cochain 11/, corresponding to the vertices of the polygon I:

'l’no....,k,,.»

Wa,,...,.-,.,

¢;,,.,.,J,,.~

(23.14)

The relation 6m¢=¢ implies that the coefficient of the cochain ¢ corresponding to an arbitrary segment of the polygon l is equal to the algebraic sum of the two neighboring distributions in (23) corresponding to the vertices of this segment. Since each coefficient of the cochain 4) belongs to the space 11,,' [J‘(R")]'"', and the first term in (23) is equal to zero because of the choice of k0, , km, all the terms in the sequence (23) also belong to this space, whence l/Iio~~~~~ hep" [J’(R")]"". We now choose some convex region “.50."..i Uhn n Ujm, containing the support of 1/110, M. The results of § 8 imply die equation

[6* (19,-, ..,j ]‘"‘ ” I" P... [6*(R")]'”' = p. [4‘ (win.....MT“By what we have already proved, the distribution 111,0”. 1," belongs to the left side, and therefore, belongs to the right side also, that is Wig, ...,j,,.=pn| ljn, my...

Where

Xi“, ...,j,.e[5'(wjo, ....j,,,)]""~

Thus, 111=p,,, x, where 1 is the ccchain whose coefficients are the distributions 11a. 1...‘ Since by hypothesis supp was, we know that for a suitable choice of the regions (him ",1... we can cause supp x to belong to S also. Then t/IEp_ ["J§(U)]"", which implies that the right column of diagram (20) is exact. Applying Theorem 1, §2, ChapterI to diagram (20), we find that the topmost row is exact. I] We now prove the theorem. Let i be a distribution in Q\£, coinciding with u in fl\(K’ u )3), where K'c Q is a compact. The distribution pl}, obviously, belongs to the kernel of the operator pl in (18). Since the sequence (18) is exact, it follows that pfi=p v for some distribution vew‘m, Z)]‘. The distribution fi—v is the desired continuation of u. Let us now suppose that M is an elliptic module, that is, p is an elliptic operator and the region Q is such that the set R"\(Qu£) has no bounded connected components. In the sequence (18), we set EZ=R". In view of our lemma, the sequence so obtained is exact. The distribution [1 :2 belongs to the kernel of the operator p1 of this sequence. Therefore p 12 = p v, where 1:6[6‘ (R', Z)]‘. The fact that

supppv=supppficK’\£, implies that supp c"\£, where K” is a compact belonging to D, since the function u is analytic in R'\(K"’UZ'), where K‘” is a compact

§l4. The extension of solutions of homogeneous systems

409

in R". It follows that the distribution fi—v is the desired continuation of u. [I

Theorem 5. The assertion of Theorem4 is valid under the following hypotheses: n=2m, [I is a holomorphy domain in C’"=R", and M is a holomorphic y-module (see § 12), and Ext‘(M,.?)=0, i =1, ..., m. Proof. Let (4.7) be a free resolution of the module M, consisting of 9”-matrices. We establish the exactness of the sequence (22). To do this, we consider the regions Uj=(V,.xL)n[l. Since each of the regions iL is convex, it is a domain of holomorphy. Therefore, the intersection (VI-XL)(‘\ Q is also a domain of holomorphy. Thus for arbitrary jo, ...,jv the region a): Uion n Ujv is also a domain of holomorphy, and therefore, by Corollary 3, § 12, the 9”-module d"(m) is flat. This implies the exactness of the sequence (22). The remaining arguments in the proof of Theorem 4 apply without change. I] 8°. The '1 of the ‘ of the ' ’ ot a ' system. Making use of the method applied in Theorems 3 and 4 we now the uniqueness for conditions sufficient obtain two theorems which yield of solutions of the system (1.4). Theorem 6. Let L be a subspace in R”, of dimension n—m, 0§mM,,, and M —>M,. Since p’fl”= 9,, n pv, the intersection of the kernels of these mappings is equal to zero. Therefore Corollary 5, § 8 implies that every distribution 14 which satisfies (1.4) in the convex region (AZ, can be represented as a sum uH+uv, where uflegfifl(9\2) and uvegfivmvi). Since the module M" is hypoelliptic, the distribution u" is differentiable in Q\Z. Accordingly, the distribution u, = u — u” is differentiable in Q\(K u 2). Thus, our task is reduced to showing that an arbitrary distribution uvefiflv (Q\Z), which is differentiable in $2\(K u E), has the same property throughout the whole region. In other words, the proof of the theorem for the module M can be reduced to the proof of the theorem for the module My, which has dimension less than n—kt In order to conserve our former notation, we shall prove the theorem for the initial module M,

supposing now that its dimension is less than n—k. We first prove a lemma. Let Yand uc Ybe arbitrary convex regions in R". We denote by a?“ Y) the subspace in 6*(1'), consisting of distributions ¢ such that sing supp 4):: 1). Lemma. For an arbitrary 9-matrix p

Wm?nptflmr=ptflmr

416

VIII. Overdetermined Systems

Proof of the lemma. Let (it be an arbitrary element of the left side. We choose two strictly increasing sequences of convex compacts K,, K,” «=1, 2, ..., tending respectively to u and to If such that supp ¢cKh and sing suppcx, We further choose a function e59“, which is equal to unity in the neighborhood of Kl. Let :1: {d‘} be a Noetherian operator associated with the matrix p. The inclusion relation ¢ep [l‘fl'fl’ implies the equation

Fim=d(e—i)¢. Since suppe¢cx2, we have e¢e[9,fz]' for some integer q. I-Ienoe, €35[S:;]‘ and, therefore, Lie—oesfidp}, where K is the highest power of z appearing in the operator d‘. Therefore

|F(Z)|§C(|Z|+1)‘“+“exp(&,(-y)),

zeN(p)t

(7-15)

On the other hand, the function (e—l)¢ is differentiable, so that its Fourier transform satisfies the inequality

lie- 1) ¢|§r(1) CXPUKJ-y»,

(8.15)

where r(z) —> 0 for |z| —. 00 faster than an arbitrary power of |z|. Therefore the function F satisfies the inequality

”'1l éi’fz) exP(’x,(-J’))s

(9.15)

where r(z)—>0 for |z| —» 00 faster than an arbitrary power of |z|. We now show that the inequalities (7) and (9) imply an inequality of the form IF(Z)I§RfZ)eXP(&.(—Y», (1015) where R(z)—>0 as |z|—>oo also faster than an arbitrary power of |z|. 1 1

In fact. suppose that lyl $271“

W,

where l=sup(|§|. CEK4}+

surtlél, CeKzl- Then exp(Ju(—y)—JK,(—y)); r’(z) and, accordingly, by (9)

|F(2)|§1/r’(2)exp(fx.(—y))We now suppose that |y| ;% ln % Then exp(—a |y|)§ [r’(z)]"", where s=§ p(x3, CK.) and, accordingly, (7) implies that

WINE C(IZI +1)“+" exp(—e|y|)exp(.);.(—y))

é C(IZI +1)"” r’(z)"“ CXP(-£.(—y))With this we have proved the inequality (10).

§15. The influence of boundary values

417

. This inequality in turn proves that FeS’,?{p}. We construct an increasing sequence of functions R42), 01:1, 2, ..., which satisfy the conditions of Proposition 5, §3, and such that r(z)+R(z)§R,(z). Since the family no; {SE} corresponds to a family of majorants .1! of type J, we may apply Theorem 2, §5, Chapter IV. This theorem shows that F=d¢, where I/IE[S§_]' for some or. Hence d[11—e +lh] =0. The inequality (8) allows us to conclude that the function l—e) (1) +1]: belongs to [SE ‘. Therefore the theorem just cited implies that

fi¢+¢=pb

1635:]

for some [3. In a similar fashion, we show that d [a—l/l] :0, where {dz—IhEBET for some q. Applying our theorem again to the family ofspaces (Si, «=1, 2, ...}, we find that

fit—w=pw,

wEES'iT.

Carrying out the inverse Fourier transform, we obtain

¢=(E¢-t5)+[(1-e) ¢+ll73=17(5c+€1), 26[9(Y)]‘, 0'16 5"(v)]’-

Heme (1)6113“ 1")J‘- I] Corollary 2. The space 6“ (T), when endowed with the discrete topology, is a flat 9-module. The proof is left to the reader.

4". Completion of the proof of Theorem 3. By hypothesis the dimension of the module M is less than n— k. Therefore by Corollary 1, § 13 we have the equation (11.15) Ext5(M,?)=0,

i=0,...,k.

Inductive proposition. Let us suppose that the module M satisfies the conditions (1 1)for i=1 . .,.k Let Tbea convex region, and let 2;, i=0.. .,,k be open halfspuces m R. Then for an arbitrary distribution ¢ep[§’* (R")]‘ with support in T difl'erentiable in the neighborhood of R"\Z'(Z’= U 23:), we can find a distribution W with support in T, which is diflerentiable in the neighborhood ofR"\):’, and such that 47: p III. This assertion will be proved by induction on k. For k=0 it follows from our lemma, if we put 0: Th 20 We shall show it for an arbitrary kzl, on the assumption that it has already been proved for k— 1. We choose a free resolution (4.7) of the module M. The module M1 =Coker p’1 satisfies the relations

Ext‘fMl, 9):Ext'+‘(M, 9)=0,

i=1, ..., k- 1,

4I8

VIII. Overdeterrnined Systems

and therefore the operator Pi satisfies the inductive proposition when k is replaced by k— 1. Let :15 be an arbitrary distribution satisfying the conditions of the inductive proposition. We choose a closed half-space 23:21, such that the distribution :17 is differentiable in the neighborhood of k

R"\(E8 U 2'),

where f: U 2,5. 1

We choose a function steak"), equal to unity in the neighborhood of R"\Z° and zero in the neighborhood of Z”. The product I! d) is differentiable in the neighborhood of R'\f. It follows that the distribution 111014» is also differentiable in the neighborhood R"\Z, and that its support belongs to (Zo\25)n Y. Consequently the distribution plump satisfies the induction hypothesis, with k—l in place of k, pl in place of p, and the region (2‘0\£5)n fin place of Y: Therefore, we can find a distribution 1/1 with suugport in (Zo\}:(,') n Y, which IS differentiable in the neighborhood of R”\ such that pl «:15: —p1|p Since the module 6‘ (R‘) is flat, the equation Ext1(M, 9)=0 implies that the sequence

[armor—H 6*(R")]'i» [6*(R")]“-

(12.15)

is algebraically exact. Since p,(a dz — 11’) = O, the distribution a: 43 —— dI’ belongs to the kernel of the operator 111 in this sequence and therefore can be written in the form

a¢—¢’=px.

16[6*(R")]‘.

(13.15)

Since the distribution aria—W is differentiable in the neighborhood of R"\Z‘, and its support belongs to R23, we can apply the induction hypothesis to it, with k—l in place of k. It follows that the distribution x inA(13) can be chosen to be differentiable in the neighborhood of R"\Z and to have a support belonging to IRES. The equation pl¢=0 implies that p,[(l—az)¢+dl']=0. Since the support of the distribution (1—0:) 4J+lpl belongs to the convex region I‘m 23, we have (1 —oz)¢+vll’=pw, me[&*(1fn£;,)]‘. (14.15) The existence of a distribution (1) with these properties follows from the exactness of the sequence (12), in which R'I is replaced by Yn 21,. Combining (13) and (14), we obtain

¢=p(z+w). The distribution 1/; = x+w is differentiable in the neighborhood of R"\Z’. and its support belongs to Y; it is thus the distribution we are seeking. This completes the proof of the inductive proposition. I]

§15. The influence of boundary values

419

Let us now turn to the proof of the theorem itself. Let u be a distribution satisfying (1.4) in (A): and differentiable 1n {2\(K UL"). For every i=0. .,k we choose an arbitrary open half--spaoe 25:); and we write 2': UL]. We further choose a function llama), equal to unity in the neighborhood of K. The distribution pfiu belongs to p[d’*(£2)]' and is differentiable in the neighborhood of R"\)J’. Applying the inductive proposition to it with Y=Q, we arrive at a distribution upe[, 259 ¢.,E 291

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