Blowing Up Grassmannians

Table of contents :
[Queen's Papers in Pure and Applied Mathematics 59]Blowing Up Grassmannians, Ari Babakhanian, 1981, 77p, Queen's University
Table of Contents
Introduction
Chapter 1. Grassmannian Spaces
1.
2. The structure sheaves on Grassmannians.
Chapter 2. Subvarieties, Segre Embeddings, Blowing up
2.1. Subvarieties of Grass(n,r).
2.2. Derivations.
2.3. Segre Embeddings.
2.4. Blowing up.
Chapter 3. Applications
3.1. Blowing up (continued).
3.2. Formal functions.
Bibliography
[1] Babakhanian, Ari; Hironka, Heisuke. Formal functions over Grassmannians. Illinois J. Math. 26 (1982), no. 2, 201-211.
[2] J. L. Coolidge, The meaning of plücker’s equations for a real curve. Rendiconti del Circolo Matematico di Palermo 40(1) (1915), 211-216.
[6] R. Hartshorne. Cohomological dimension of algebraic varieties, Ann. Math. NO. 4 (1968), 587-602.
[9] H. Hironaka. Resolution of singularities of an algebraic variety over a field of characteristic zero, Ann. Math. 79(1964), 109-326.
Annals of Mathematics Volume 79 issue 1 1964 [doi 10.2307_1970486] Heisuke Hironaka -- Resolution of Singularities of an Algebraic Variety Over a Field of Characteristic Zero- I
Annals of Mathematics Volume 79 issue 2 1964 [doi 10.2307_1970547] Heisuke Hironaka -- Resolution of Singularities of an Algebraic Variety Over a Field of Characteristic Zero- II
[10] H. Hironaka. On some formal imbeddings, Illinois J. Math. Vol. 12, No. 4 (1968), 587-602.
[11] H. Hironaka, H. Matsumura. Formal functions and formal imbeddings, J. Math. Soc. Japan, Vol. 20, Nos. 1-2 (1968), 52-82.
Introduction.
\S 1. The ring of formal-rational ...
\S 2. The effect of proper ...
THEOREM (2.1). ...
THEOREM (2.6). ...
\S 3. Embeddings into ...
\S 4. Embeddings into ...
\S 5. Examples (the case ...
Bibliography
[13] S. Kleiman, J. Landolfi. Geometry and Deformation of Special Schubert Varieties, Compositio Mathematica, Vol. 23 (1971) 407-434.
[16] 0. Zariski. A simple analytic proof of a fundamental property of birational transformations, Proc. Nat. Acad. Sci. USA 35(1949), 62-66.

Citation preview

BLOWING UP GRASSMANNIANS By Ari Babakhanian

Queen's Papers in Pure and Applied Mathematics - No. 59 Queen's University Kingston, Ontario, Canada . .

1981

COPYRIGHT

@

1979

This book or parts thereof, may not be reproduced in any from without written permission from the author.

TABLE OF CONTENTS

Introduction

1

Chapter 1:

Gras~mannian Spaces

21

Chapter 2:

SUbvarieties, Segre Embeddings,

35

Blowing up Chapter 3:

Applications

Bibliography

49 75

INTRODUCTION

This work contains the theory of monoidal transformations, or blow-ups, of Grassmannian varieties. Since our constructions are novel, we have included an application of our theory to the problem of the determination of the function field of formal functions along subvarieties of Grassmannians. a substantial part of Chapter 3.

This occupies

Classically these

constructions were carried out with the aid of the Plilcker imbedding of the Grassmannian space into an appropriate projective space, thereby utilizing the homogeneous coordinate system of the projective space . . A careful examination of the classical constructions of monoidal transforms reveals that it is the ~uotient line bundle

0(1)

which plays a crucial~ole.

We have

exploited this phenomenon here by constructing monoidal transforms with centers on Grassmannian varieties utilizing the quotient vector bundle.

The use of the

quotient vector bundle proves to be essential in the application we have carried out in Chapter 3.

Our

method, in fact, grew out of the effort to compute the function field of formal functions along a projective

2

IP 1

line

in a Grassmannian

method of imbedding

Grass(n,r).

Grass(n,r)

The classical

via the Plucker map did

not prove fruitful, and led to our consideration of the problem in a monoidal transform of 3

Grass(n,r).

Chapter

contains the detail of this monoidal transformation

and some computations in the blow-up which is analogous to work done by Hironaka-Matsumura in the projective case. I am grateful to Heisuke Hironaka for his help in the formulation of the ideas here.

Thanks are also

due to Hilda Britt for her superb typing of the manuscript.

In order to give the motivation for the constructions of the monoidal transformations in chapters 2

and

3,

we begin here a short treatment of monoidal

transformations of projective varieties, and specifically of projective spaces. Z

Broadly speaking, given a scheme

and a closed proper subset

X

ested in constructing a scheme

of

z'

we are inter-

and a proper

birational morphism

f : Z ' --> Z

striction of

f

Z1 - f- 1 (X)

dimension of

r- 1 {x)

to

Z,

such that the re-

is one to one, and the

is greater than the dimension of I

X.

We will say

Z

is blown up by

is the center of the blowing up.

f

to

Z ,

and

X

3

Before giving the definition of blowing up, we will consider some examples.

Example

Let

1.

Z

=

JPn

CX 0 ,

homogeneous coordinate·s subvariety of

x 2 =o. X.

IPn

be the projective space with •••

Let

,.xn).

defined by the equations

We will construct a blow up of IP 2

Let

X

IPn

be the

x0 =o, x1 =O, with center

be the projective space with homog~neous ,.{

coordinates

(Y 0 ,Y 1 !Y 2 ).

Consider the projection

p:IP n -X-->IP 2 p Cx 0 , •.• ,xn )_ = (,x 0 ~x 1 ,.x 2 L

defined by

p

is surj ecti ve

and can be extended to a correspondence

c IP n x JP 2 .

Z

I

is an extension of

. graph of

Ee X,

p

rr

p

since it is the union of the

with the points is any point of

(E;rr), IP 2 .

is blown up by the correspondence JP 2 .

Let

projection

f : Z' --> Z

where for each

Thus each point of Z

I

X

to the whole of

be the map induced by the first

4 I

f : Z --> Z of

z

in

z'

z.,

is blowing up (or monoidal transformation)

with center

x.

To examine the coverings via IPn -

of the open covers

we observe that

z

r

f

V(X 1 ), i = 0, ... ,n

of

contains the open affines: 0 < i < 2.

V . . = z' n (JPn- V(X 1.))x(IP 2 - V(YJ.)), 3~i 2 •

jective variety isomorphic to A

f-l (X)

somewhat simpler and well known example is

blowing up of a point on

TI'n,

which we will construct

ne.xt (see Hatrshorne, Chapter 1, §4 [ 7 J,

Hironaka-

Matsumura Lemma 3.1[11],

8 §8A[l4]):

Examp 1 e

2.

Let

Mumford Chapter

=

Z -- TDn

homogeneous coordinates

be the projective space with (x 0 , ••• ,Xn).

Let

X be the

subvariety of dimension zero consisting of the single point

x = (O ••• ~,O,l},

i.e.,

defined by the equations will construct a blow- up of JPn-l

X is the variety in X

Q

JPn

=O We n-1 · with center X. Let = 0

'· • • '

X

be the proj ectiye space with homogeneous

coordinates

(Y 0 , .•. ,Yn-l).

Consider the projection

p : JPn _ X - - > TI'n-1 defined by

p

is surjectiye and can be extended to a correspondence

7

z'

p

is indeed an extension of

of the graph of

where

1T

since it is the union

p

and the points ((0, .•. ,0,1). ; TI')., is any point of JP n"""l . Thus the point

is blown up by the correspondence z ' to the whole of JP n-1 . As in Example 1 we will e.xamine

co' ... '0 '11

the coverings in z ' o.f the open affine covers of n Let f : Z'· -.-> JPn be the map induced by the Z=F • first projection

f : Z' --> JPn

is blowing up (or monoidal transfor-

rnat;:tonl of

with center

JPn

th.e coverings via

f

Z'

in

ll?n ... V(X1 1., i=O, ••• .,n

of

(0, ... , 0 .,1).

To examine

of the open covers JPn

Z'

we observe that

contains the open affines: u· = z ' n (_JPn i

vcxi 11

X

(JPn-1

V(Yi).)., 0 -< i < n-1

z ' n (_JPn

vcxn).).

X

CJPn-1

VCY j ).)_, 0 < j < n.:..1.

yj =

f : z ' -.->. IPn

al

Under the projection

ui

is mapped isomorphically to the affine

the affine JPn

-

V(.Xi),

i = O, •.. ,n-1.

for b)_

'

The affine coordinates in the ambient affine space

K2n-l

containing

Vj

are

8

X

XO

xn ' ... ., zn... 1

n-1 -x'

zo

=

WO

= v-: ' ... ' wn-1 =

;::

n

YO J

Yn-1

½

Z' = V(. •. ,Xi Yj-XjY 1 , .•. ),

It follows, from the fact that

that in these coordinates the affine the

n-1

omitted). ' (Wj

is given by

Vj

equations ,.

Cll

z

so that

vj

a

=

z.w' J a

-= Kn.

we get

C2 )_

= o., ••• ,j, ... ,n-1,

a.

Solving the equations

wa

for

(l)

z

a wa = z·

j

fhe affine

Vj

with generic point ,.

(ZO., ••• ,Zn-1;

z0

_J_

J

J

Z.

z:-,· .. 'z.,••

is isomorphic to the affine space coordinates

Z

l

n-) •,-z:-J

Kn

with affine

,.

(31

(Zj,

z0

Z.

J

J

Z

l

_J_ n-) z:-,·•·,z.,•··,-z:--·

J

It follows that the function field of isomorphic to the function field of · JPP

and so

t

Z

is

f

is

birational. Th_e dilllens ion _of { (0., • • • , 0, 1)}

'X

J!.>n,.._l

:ts

f

-1

n-1,

..

CLO ., • • • , o ., 1 ) )

=

Since f{O, ... ,O,ll}

is

9

vc.z 0 , ... ,zn_ 1 ),

the variety f

-1 ·.

n V.

(CO, •.• , 0, 1}}

= V(Z. J J

it follows that ~

zo z, ... ,z., .... ,Z. j J J

•

z

n-1 1 =

z.

J

vczJ l.

r -1 cc.0, •.. ,0,1}) nv.

Therefore

J

Kn -= Vj

subspace of

is the af.fine

with affine coordinates the subset

A

zo ~ ·.· .·. 2 11-1 of Z. , .• ~,z- · , ... , z· j

j

n

f

K •

-1 .·

JPn-l Example

the affine coordinates

(JO, ••• ,Q,l)_),

with homogeneous coordinates Let . Z = !P 2

3.

bmic·· set in

X

C

in

with cente;r,

2

JPX

is the set

C0,0.,1)_

of

in total, is the projective· space

Let

c

be the a1g·e ....

0

JP_i.•

the three point

(_l,O,O).,

(.0,1,0}.

We will construct a blow up of Let

JP2

homogeneous coordinates

defined by

be the projective space with

. given by the equations

bf

c.

(Z 0 , ••• ,zn-l).

cx 0 ,x1 ,x 2 l.

ha,mogeneous coordinates

c

(3)

j

y

and

IP_i ·

be the projective space with CY0 ,Y 1 .,Y2 ).

Consider the map

p(x 0 ,.x1 ,x 2 l = (.;x1 .x 2 ,.x 0 x 2 ,x 0 x 1 ).

e.xtended to a_·. correspondence

2 z' c JP X2 x J1> . y

p

can .be which is

10

given over the affine

z'

2 (JP2 - y - X - V(X 0 l) x(IP

()

-

V(Y 1 ))

by

Yo yl

x1x2 xl = XOX2 XO

=

y2 XOXl X1/Xo = = yl XOX2 X2/Xo So

z

I

is the variety of the equations

in the Segre imbedding s

JF·2

X

X

IP2 - > IP8 Y

Z

We I

will show that

Z

is an extension of

the union of the graph of

p

p,

with the sets

in that it is ((l,O,O),

(~,Y1,Y2ll, ((O,l,O),(Yo,o,Y2)),((0,0,l),(Yo,Y1,0)) points of

2 IPX

Let

of

2 IPy.

x

f : z ' -->

first projection

be the map induced by the

11

2

]PX

We will show

f

co,1,01.,co,o,1}

.

X

.. 2

2

]Py -.-·-> n>X.

blows up only the points (1,0,0},

of

2 JPx,

each to a projective line in

z' .

X

f'

first

projection

12

(I)

'I'he open set

has affine coordinates (with respect to the covering by

, with relations

' Thus

f

Z' -->

one map onto

IPi

- V(X 1 X2 ).

a one to

Similarly let

then the monoidal transformation

u1 Cresp. u2 )

0 is

u

restricted to

f

restricted to

is a one to one map onto

IPi - V(X0x2 ) (resp. IPi - V(X0X1 )).

13 (IIl

The open set

(.Cz 1 ., 0 .,z 2 ., 0 }.,(.W0 ., 1 ,w 2 ., 1})

has affine coordinates zl,O =

w0,1

x1

X' 0 Yo yl '

z 2.,0 =

x2 ., XO

y2

w2 ; 1 = yl .

and ·relations Z-1,0 -- = WO ., 1·

Similarly_

Thus

v0 , 2 ,has affine coordinates

r-1(-.cl' .

n

v0

'

1

has affine coordinates

((O.,O),(O.,W 2 , 1 )) and

f- -l(-Cl . n Vo, 2

has affine coordinates

with

14

cco ,ol , co ,w1 , 2 ) ) with relations

Thus

on 1) Z' n

r- 1 (c) n (v 0 , 1 u v 0 , 2 )

JP.i

x II'~

is the projective line in

consisting of the points

Similarly: 2}

f- 1 Cc) n (Vl,O

U

v1 , 2 )

is the projective line con-

sisting of the points

(JO, 1, 0), (YO, 0, Y2 ))

3}

f- 1 (c) n (V 2 ,0 U

v2 , 1 )

C

JP.i

X

1Pi ,

is the projective line con-

sisting of the points

We will now begin our general discussion of blowing up on a projective space. cal line bundle

O n (1) JP

on

lPn.

Let Let

U

be the canoni(x 0 , ..• ,Xn)

be

15

the homogeneous coordinates on

JPn.

The line bundle

is generated over the sheaf of rings O

U

by the global

lPn

sections

Over the covering by the

n+l

Jpn - V(Xi}, i=O, 1, ... ,n., bundle

U

Cal U

is generated by

open affines

the patching data far the

are as follows:

[X1]

over the affine

JPn - V CX1 }, Cb)_

X.

- :t.- [Xi],

j=O., •••

.,n,

l

over the affine

JPn - V(Xi} .

we define

u®v

to be the line bundle over

. generated over the sheaf of rings

by the global

sections

Cl! 1 where

1 1 , ... .,iv

]Pn - vcxi}_ (5)

[X1J

E

the line bundle @ • • •

Over the covering u8 V is generated by

{O,l, •.• .,n}.

@

[X1] (.v ... fold tensor product)

where we identify the product

C5}

with the symbol

ll?n

16 \)

[Xi].

Then for the glohal section JPn - vcxi)

datum over

is

xi

[X. J

(61

®

ll

C

C

8

0

[Xi J =

• • • X.

1\)

1

A of

f O, ... , f N e

. global sect ions braic set V(A)

of JPn

xi

U8v

generated by the

r (Jpn, u9")

defines an alge-

by

V(A) = {x e JPn jfi(xl=O,

The global section

[X~].

\)

\)

A subbundle

the patching

( 4).

f O', ,a=l, ... ,N

f 1 e A}.

over the affine

can be written (7)

(6}.

with the aid of

Suppose the global sections

r 0 , ... ,fr

e

u8 "

\)

are s-uch that:

t 0 , • •. ,fr

(Ll

are linearly independent over

K.

\)

CI:I 1

on

If

f

is a vector in

VC! 0_, ... ,fr ) , \)

tha,t

r( JPn, u19 "),

then there are

c0,

and vanishes

..• ,er e K such \)

17 Consider the projective space coordinates are

Yo' . • . 'y

r"

JP~

whose homogeneous

Let

.

x = (.x 0 , •• • ,xn) e JPn r (ip 0 (:x1, ••. '"°r (x)) e JP " where the homo-

be the map that carries the point to the point

geneous polynomial The map

p

"

.

is as defined in (7).

ipa(x 0 , ••. ,Xn).

can be extended to the correspondence

where

z'

vc. .. ,r1

=

8 [YJJ -

rj

0 [Yi], ••• ),

o~

1, j ~ r".

e· [Yj] - fj 8 [Yi] is a global section of' the line bundle u8" 8 W over the projective variety

Here

fi

where

'

W is the canonical line bundle

generated by the global sections

r

JP "

Let

f : z' --> lPn

first projection

lPn x lPr"

[Y 0 J, •• ~, [Y

r"

J

over

be the map induced by the

--> :lPn •

f : Z ' - > JP n

is called the blowi~g up (.©r mo.noidal transformation) of

JPn

with center Under

V(f 0 , .•. ,fr )..

"

f : Z' --> JPn, ..

the affine open set

18 is mapped isomorphically onto the affine

r" u u1

and

Z•

in

is isomorphic, via

lPn - V(f 1 ),

f,

to the open

i=O

lPn - V(r 0 , ••• ,fr 1 of

subset

lPn.

V

The affine coordinates in the ambient space n+r K

containing

"

( (Z o , i , ... , Zi ' i , . . . ' Zn , i ) ' (.WO , j , . • . , W j ' j , • .. 'W r v ' j ) ) •

are

xa

where

za,i = X1 '

Wa,J

y = y~·

In these coordinates

v1 ,j

is defined by the equations

wB,j

( 8).

where of

v 1 ,j

affines of

"'a'

JPn

"'J is

are as given in n.

Thus

f

(7),

so that the dimension

maps each one of the

rv + l

isomorphically to the affine with coordinate A

(Zo ,1., ... ,z.1, i, ... ,z n, 1>· To examine

we first observe

19

V(f O, ... , fr ) n (Jpn - V(Xi})

that

is the variety

\)

V (IP O. CZ O ,i• ' •••

.,1 , ... ' zn,i. L ... 'cp rv CZ O,i• , ••• '1 '

Abbreviating

cp µQ,,

••• ' zn,1.. ) )_.

cz O,. i• , • • • , Z1·-1 ,1• , l , Z.i +l ,1.. , • • ~ , Zn,1• l

by

the af .fine

cp 13

f

-l''

·.··, '

Cf O, • • • , fr ) n V • . \) 1.,J

= V(v> j l. f -1 (f0

Thus

, •••

) n Viij

,fr

is the variety of

cpj

=O

\)

togethe;r, with the

rv

identities

(8)

n +

in the '

r \)

coordinates ~

Zo·.·

• , ••• ,

,1

~

i, w0.· • , ••• , W. . ., •• ,. , w: .• n, ·:,J J,J rv,J

Z·, .. , ••• , Z

1.,1.

It follows that

{- 1

erO, ••• , fr

VCf 0 , ••• ,fr } n OPn - VCXi1), '

) n Vi J',

'

\)

'

the blow up Of

is of dimension

n-1.

\)

Finally one cart define the blowing up of a Noetherian scheme with. center a closed subschetne as follows: Z

b.e a Noetherian scheme, and

c6rrespohdihi to a c6herer1t

C

shear

Let

a closed sub scheme of ideals

I

on

x.

20 Cl).

G=

Consider the sheaf let

Z ' = Proj G.

l

i=O

Ii

of_graded algebras and

Then the canonical map

is· th.e blowing up of

Z with center

Zt --> Z

C.

The constructions of blowing up of Grassmannians could be carried out using the general definition of blowing up of Noetherian schemes.

This,

however, would involve imbedding Grassmannians via the Plllcker map into a projective space and constructing the monoidal transformation with the aid of the quotient line bundle on the ambient projective space, thus making the study of the blowing up of the Grassmannian (.as a subvariety of

JPN)

cumbersome.

In this work we have

utilized the quotient vector bundle on (our ambient space) the Grassmannian for constructing the monoidal transformation.

This makes possible the analysis of the blow up

· carried out in Chapter

3.

21

Chapter 1 Grassmannian Spaces

1.

Recall that if

K is a field and

vector space of dimension

r

Ar

the affine

with basis vectors

e 1 =(l,O, ... ,O), e 2 = (o,1,0., ... ,o), .•.. ,eri= (O, ... ,O,l), we define the projective space IPKr-'l as follows: for r any non-zero vector V = xiei consider the orbit r=l Q(v) = {sv : s E K - {O}} of V under the action of

I

K* = K

IPr-1 (or K be the space of these orbits, i.e~,

-

{O}.

We define

Grass(l,r))

to

IP~-l = (Grass(l,r)=)

Ar - {O}/K*.

To

define the Grassmannian space

we follow the above procedure with

n

independent vectors

Ar.

Q(v 1 , ... ,vn)

v 1 , ... ,vn

in

Grass(n,r),

linearly Let

be the orbit under the action of

GL(n,K)

on then-tuple of the linearly independent vectors (v 1 ,.~ .,vn)

as follows:

If

22

n(v 1 , ... ,vn)

is, in fact, the collection of all bases

which generate the subspace v 1 , ... ,vn. subspace

Thus An

struct the

An

n(v 1 , ... ,vn)

of

Ar

is synonymous with the v 1 , ... ,vn.

generated by the vectors n x r

matrices

generated by

(Kij)

Con-

using the coeffi-

r cients in the

n

identities

r

Vi=

Kijej,

j=l i = l, ... ,n. rank

n.

on the

is the quotient modulo the action n

x

r

matrices

We note that the space matrices fold of

are a fortiori of

Grass(n,r) to be the space of these orbits;

Grass(n,r)

GL(n,K)

(Kij)

n(v 1 , ... ,vn) = {a(Kij):a e GL(n,K)}.

Then

We define i.e.,

These matrices

of rank

( Kij) n

n,

Ar ,

frames in

n

the affine space

X

r

of rank

St(n,r)

of

n•r

n x r

is the Zariskl open comn

X

n

subdetermi-

matrices of rank less than

A

n.

called the Stlefel mani-

plement of the set of zeros of the nants of the

(Kij)

of the matrices

n

in

Thus

23

Grass(n,r) action of

is the quotient of

modulo the left

GL(n,K).

Example 1. n(v)

St(n,r)

If

V

is a non-zero vector in

is the collection of all vectors

KE K,K i 0.

So

Grass(l,3)

KV

then with

is synonymous with the A3 .

collection of all the lines through the origin in Hence

Grass(l,3) = JP 2 .

Example 2. then

If

v1 , v2

n(v 1 ,v 2 )

is synonymous with the plane, in

through the origin which contains the vectors v2 .

Hence

Grass(2,3)

varieties.

A3 , and

is synonymous with the collec-

tion of all the planes through the origin in Proposition 1.1.

A3 ,

are linearly independent in

A3 .

Grassmannian spaces are projective

To show this we will construct an ,imbedding,

called the Plucker imbedding of

Grass(n,r)

into

(r)-1 (K .. ) E St(n,r) to the point in lPn with lJ coordinate the determinant of the i 1 , ... ,in-th i 1 , ... ,in-th

columns of

(Kij).

This map induces

(r)-1 1r

Grass(n,r)

-+

IP n

(called the Plucker map)

24

since for any

cr e GL(n,K), o • (K 1 j)

i 1 , ... ,in-th

point with

entry the constant

times the determinant of the ( Kij).

Thus

point of

( Kij)

and o(Kij)

n

be the points in

i 1 , ... ,in.

identity matrix. Grass(n,r), (r)-1 lP n

TI

whose

of

(Kij)

imbedding.

have linearly inde-

i 1 , ... ,in

(Kij)

columns form the

u

11,···,in

(K. ) e U ij i 1 , ... , in i 1 , ... , i n-th

" i 1 , ... ,ii,···,in,in+j-th

11,· .• ,in

Wewill represent

The the sets maps

u

whose representative

(Kij) e St(n,r),

matrices

suitably so that the

n

columns of

are mapped to the same

is an imbedding, let

Grass(n,r)

pendent columns

l.

1 1 , ... ,in-th

det o

(n)-1 lP r . . To show

n x r

is mapped to a

with suitable parity.

Thus

cover to the point

en t ry i s

entry is the

n x n

1

and the

(1,n + j) TI

entry

is an

In fact, we can say more:

Proposition 1.2.

The Grassmannian space

Grass(n,r)

is a non-singular, irreducible projective variety, and the Plucker map

7T

Grass(n,r)

+

( r)-1 lP n

25

(r)-1 is an imbedding whose image in

IP n

is the pro-

jective variety of the homogeneous ideal generated by the quadratic forms

Ri 1 ... i ;j 1 , ... ,j ;m = det(K. . )det(K. . . ) n n 11' ... in J l' ... ,J n n

+

l

det ( K.

•

•

•

•

)

1 1··· 1 m-1Ji 1 m+1··· 1 n

i=l

To show the image of

Grass(n,r)

under the

Plucker map is the projective variety of the homogeneous forms of

R.

1

.

.

.

1· · · 1 n;J1,· · · ,Jn,m, det(K.

. ),

11 ... ln

let Then

Ak

be the cofactor

26

X

det(K. .

+

(-1)

.

.

, )

inf 1 · · .J £-lJ £+ l" · ·Jn

n-m

n

n

l

l

,

,

Q, k ) (-1) A Kkj det(Ki , . j . k=l t=l Q, rrfl · .. J t-1 t+l .. ·Jn

Kki

m

n

= (-l)n-m I

Ak • K11

k=l

m

Kkj ••• Kkj Kkj ... Kkj 1 t-1 t+l n Klj

1

Klj

t-1

Klj

t+l

••• Klj

n

=0

since the determinant in ea.ch term of the summation has two equal rows.

27

(K

Examples.

under the

Denote the image detl li K2i

(r+l)(r-2) Plucker map

TI

:

Grass(2,r)

(a)

The image of

TI

(b)

The image of

TI

:

+

2

F

by

X •.•

lJ

Grass(2,4)

-+

F 5

is the hyper-

Grass(2,5)

+

F 9

is the pro-

jective variety defined by the three equations

Denote the image

X12X34 = X13X24

X23X14

X12X35 = X13X25

X23X15

Xl2X45 = X14X25

X24X15

Kli

Klj

det K 2 i

K2 j

K3i

map by (c)

K

under the Plilcker

3j

xijk'

The image of

TI

:

Grass(3,6)

+

F 19

is the pro-

jective variety defined by ten equations:

nine

quadratic equations, each with three terms, e.g., X123X145 = Xl24Xl35 - X134Xl25'

and one quadratic

equation with four terms, namely the equation

28

2.

The structure sheaves on Grassmannians. 0

section we define the structure sheaf the universal quatient bundle sheaf

D on

visorial bundle sheaf

0

D

=

IP r- 1

(D)

n = 1

For

O ,0 (1) Irr- 1 IP r- 1

and

where the homogeneous coordiate

i

O

ring of

and the di-

associated with the divisor

{((O .· r-l)X ,Xi)} IP

Grass(n,r)'

Grass(n,r).

these are respectively the sheaves the sheaf

U

In this

is

IP r-1

K[X 0 , ••• ,Xr].

OGrass ( n,r ) ·

The strusture ring -

Let

X

be

(r)-1 the image of

1r

:

Grass(n,r)

IP n

-+

Then

the variety of the homogeneous prime ideal

X

generated

P

by the quadratic forms given in Theorem 1.2.

S = K[{Ki 1 ... in}

is

The ring

]/~. is graded since

p

l 2.

JPn

'

Then the function field of the formal-rational.

functions along

in

X

tional functions on

JPn

is exactly the field of ra-

JP n.

One of the crucial steps leading to the above theorem is the homeomorphism above.

u1 u v 1

- JP 1 x An-l

proven

We will now show that this homeomorphism, and in

fact the above results on formal-functions, can be extended to Grassmannians. Consider the set represented by

n x r

P

x e Grass(n,r)

matrices

r~l°"

(1)

of points

.Klr

l

K~1 • •• Knr

described by: i 1 , •.. ,in n,

then

x e P

in the n + 1

if whenever the rank of the columns

n x r is one of

matrix (1) representing i 1 , ... ,in.

Thus

Example 3, can be used to define a projection Pn+l : Grass(n,r) - P

+

Grass(n,r-1)

P,

x as

is C

in

64

(

l

K~l ••• K~

Pn+1 :

by

:

The projection

~

:

Knl ••• K

, Pn+l

nr .

with center

P

can be extended to the correspondence

z, Z c: Grass(n,r)

Grass(n,r-1)

x

where

(2)

with

K.

.

11" .. in

the

(i 1 ... in)-th Pliicker coordinate of the

generic point

rKll • • .Klr

IlKnl ... Knr =

of

Grass(n,r),

and

n.

.

.

J1···Jn

coordinate of the generic point

the

Plucker

65

of

Grass(n,r-1),

where

i 1 , ... ,in' j 1 , ... ,jn

n + 1

in (2).

Thus the center of

P

Grass(n,r-1).

x

a variety itself.

1 . ( rn-lJ

u.

11·

by

P

is blown up by

Moreover,

We denote

call it the blow up of BP(Grass(n,r))

does not occur among

Z

Z

(r~l J

to the whole

is irreducible, hence

by

Grass(n,r)

Z

Bp(Grass(n,r)) at

[(~:i) + 1]

P.

and

We can cover

open affines as. follows:

open affines

i = ·· n

R_(Grass(n,r)) n (Grass(n,r) -V(K.

11·

-p

where

V.

n + 1

.

.

i )) x (Grass(n,r-1) -V(n1

·· n

does not occur among

.

1 1 · · · 1n ,J 1 · · · J n

1. ))

1··· n

1 1 , ... ,in.

=

BP(Grass(n,r)) n(Grass(n,r) -V(K.

i )) x (Grass(n,r-1) -V(n.

11·

where

n + 1

occurs among

among

j 1 , ... ,jn.

·· n

1 1 , ... ,in

J. ))

J1 · · · n

and does not occur

66

Under the first projection p 1 : Bp(Grass(n,r))

Grass(n,r)

+

jection of the product Grass(n,r), U.

.

.

induced by the first pro-

Grass(n,r) x Grass(n,r-1)

onto

goes isomorphically to the affine

l l ... in

Grass(n,r) - V(K. . ). il ... in

Moreover,

u

. . ) ( ll' ... ,in

u.

.

ll ... in

i ;i!n+l a

covers that part of to

BP(Grass(n,r))

which is isomorphic

Grass(n,r) - P. In order to give similar interpretation of the

open affine

V.

1

.

.

.

1··· 1 n'J1···Jn

'

the ambient space which

contains this open affine has affine coordinates:

-1

W = Tl. . •n where n + 1 kl · · · kn J 1 · · · J n kl · · · kn j's

and k's. For example, for

affine coordinates (n)

does not occUr among the

(n)

vl,n+2'''''vn,r

z

V 1, ... ,n-1,n+l;l, ..• ,n

k 1 ... kn

are

(n)

v1

,n

, ... ,v

(n)

n,n

the ,

and certain homogeneous polynomials in

67

these, where

(n)

Vi ,n+j

-1 =K

•K

l, ... ,n-1,n+l

affine coordiates

wl, ... ,fi,n+j

1, ...

~ ,1, ... ,n- 1 ,n+ 1 ,n+ j•

The

are

-1

=

n1, ... ,n • nl, ... ,fl,n+j

=

K

-1

• K

l, ... ,n

l, ... ,fi,n+j

v(n)

1. .. 0

1n --:ff{y

(n)

1. .. 0 vl,n+j

V

,n,n

= det ,:

det ·

(n) n-1,n

V

o... 1

Cn)

-:

vn n 0

V

n,n+j

1

~ n,n

(

1. .. 0

(n)

0 ... 0

'

0

(n) 0 ••• 1 vn-1,n+j

(n )

vl,n+j -

(n) (n) Vl ,nVn n+'J

1

1

v(nJ

n,n

= det 0 ... 1

/n)

(n)

vn-~,n+j

v(n)

n-1,n n,n+j v(n)

n,n

=

vf7?+J •,

vn

j = 2, ... ,r - n

n,n *Compare the computation of

w1 , ... ,n,n+J A

•

with the projective case.

68

and with similar computation

wl, ... ,i, " ... ,n,n+J.

-1 = nl, ... ,n • nl, ... ,i,., ~ ..

,n,n+j

/n) ./n) .

=

(n)

vi,n+j

1,n n,n+J v(n) n,n

for

i = l, ... ,n - 1

and

j = 2, ... ,r - n.

Thus

V1, ... ,n-1,n+l;l, ... ,n· is isomorphic to the affine space An(r-n)

with coordinates

l

r v(n).v(n) ) v(n) (11,) (n) (vl n'···, n n' vi n+j - i,n (n)n,n+jt ' ' ' ' V • • n,n i=l, ... ,n-l;J=2, ... ,r-n

{

v(n).}

:~~)j n,n

)

J. . J-2 ' ... ,r-n

More generally, for coordinates

V1, ... ,i, " ... ,n+l;l, ... ,n the affine (i) (i) (i) (i) zk , ... , k n are v 1 , i, ... ,vn,1.,v1 ,n +2 , ... ,vn,r 1

and certain homogeneous polynomials in these, where (i)

-1

v,...,u ,n+J· = Kl , coordinates

"" ... ,n+ , 1 • K1 , ... ,a, " ...· ,n,n+J. . The affine ,1, Wk k are (compare with blowing up a point

•••

1, ... , n

69

w1, ... ,fl,n+j

-1 = n1, ... ,n • n1, •.. ,fl,n+j -1 = K

l, •.. ,n

i-th colunn1

...

1 •

0

• K

l, ... ,ft,n+j i-th column

/i) l,2i 0 - )i) n,i

1 ... o /i) l.,i

•1

i-th row 0

1

1

0

vn ., i

= det

o... 0

(i)

••• 0

vl,n+j

..•

v(i) 1. •• o i,i

det

/i) 0

1

tr,

0 -

.

vn i

'

0

0

...

0

v(i~ n,1.

(i) (i) (i) vli vn)n+j vl,n+j (i vn ., i

(

1

1

/i) n-1 i ... 1 (i~ V n.,i

0

v(i) n,2n+j v(i) n.,i

i-th row

= det

ii)

1 0

= v(i)

n-1,n+j

v(i)

0

-

n-1.,n+j

'

-

v(i)

n-1,i n,n+j

(i)

vn ., i

(i)

0

•• • 0

V

n.,n+j

70

w1, ... ,i, ... ,n,n+j A

=

n-1 1, ... ,n • n1, ... ,i, ... ,n,n+j A

-1 = Kl· 1' · , ... .,n • Kl., ... ,1, ... ,n,n+J•

l

/i)

1 . . . 0 ...

- n,1 trt V

1

•

1

I

1

i-th row 0 • • • 0

0 /i) l,n+j

f1

(i) n,1•

V

det ·

= det

v~i~ 1

l

l

- )i~

n,1

(i) 1 • n)1 l (i

1 (") J 1 o ... ov n,n+··J

V

Vn,1•

J

(i) (i)

r1

. •. 0

i-th row O

1 0 0

(i)

••• 0

vl,n+j -

v1 ,1.vn)n+·J v(i n,i

0

1 = det

1

(i)

= Vn,n+·J

vCi)

n,i

(i)

.. n-l ,n+J

V

/i)

v(i)

n-1,i n,n+j V

(i)

n,1•

71

For

i < a

. .. ,a, w1 , ... ,1, " ... .,n,n+·J

-1

= nl, •.• ,n • n1, ... ,i, ... ,a, ... ,n,n+j

(

11

0

i-th row 0

0

...

O

/i) l,i

1

- ::11T

...

i-th column (i) vli

det

0

1

(i) vii 1

'

det

1

v(i) n-lyi 1 (i V . n,i J

0

0

(i) (i)

v(i) a,n+j

For

a
0 at the generic point of X, and so a is generically surjective. Letting R= ima K = coker a, we have R E C since it is a quotient of G; K e C since K must have support < X, and finallyFe: C since it is an extension

COHOMOLOGICALDIMENSION OF ALGEBRAIC VARIETIES

407

Using Serre's characterization of affineness,this proposition becomes a generalization of Chevalley's theorem [EGA, II 6.7.1], which states that if X is a finitesurjective morphism,then X' is affineif and only if X is. f: X' Note that the proposition applies to the normalization X of an integral scheme X. Thus we have -

cd X=cd

X,

q(X) = q(X) .

and

A similar argument, applied to the Leray spectral sequence of a proper morphismf: X Y, shows the following. If the fibresof f are of dimension we have r. This follows from [EGA, III 4.1.5] and [5, Th. 3.6.5]. A frequent situation of interest is when X is a proper scheme over a field, Y is a closed subset, and we wish to consider the open complement U= X- Y. If F is a coherent sheaf on X, we apply the exact sequence of local cohomology [LC, 1.9], giving -*

H'(X, F)->

H(U.

F)

-,Hyi'(X,

F)-

>H+'(X,

F)-*

Since the two outside groups are finite-dimensional,one middle group is finitedimensional if and only if the other is. Hence, since every coherent sheaf on U extends to a coherent sheaf on X, we have PROPOSITION1.2. If X is proper over k, Y is a closed subset of X, and U = X - Y, then q( U) is the smallest integer n such that Hy(X, F) is finitedimensional for all i > n + 1, and for all coherent sheaves F. Thus the integer q( U) depends only on local informationaround Y. The cohomological dimension cd (U) is a more subtle invariant, however, and depends on X globally. Let us look at some examples. Any irreducible non-complete curve is affine(see e.g. [3, Prop. 5]) so for curves the situation is very simple complete curve: q = -1 non-completecurve: q

=

0

cd = 1 cd

=

0.

For surfaces, the situation is a bit more complicated, but can be worked out using standard known results (let the reader supply the details). Suppose X is a complete non-singular surface, and let Y c X be a closed subset. We consider U = X - Y. If some irreducible component of Y is a point, then dim H'( U, Ou) = c 0

,q

=

0.

(Y2)


n-1>

cd(P-

Let F = (p(-n - 1). sequencewe have Hy",(F)

=n-2. p-i - k. From the local cohomology

Then Hn(P, F) H(P,

-

F)

H"(P

-

-

Y1, F) = 0

the zero on the rightbecause of Lichtenbaum'stheorem. Hence dim H~y(F) > 1 .

Similarly dim Hy2(F) > 1 .

Because Y1and Y2 do not meet,we have = H1 (F)

HI,2(F)

EDH"2(F)

so this k-vectorspace has dimension> 2. The local cohomologysequence for Y1 U Y2gives ***

> H--1(P

- Y1 U Y2, F)

H1l~12(F)

-

-

Hn(P, F)

-

0

so that Hn-l(P - Y

U Y2, F) # O

as required. The next propositiongives anothertechniqueforcalculatingsome cohomologicaldimensions. PROPOSITION1.4. Let X be a scheme, and let Y be a closed subscheme. Assume

(1) cd Y = 1

(2) Hy(F) = 0 for all i > m, for all F coherent on X (3) cd (X - Y) (resp. q(X - Y)) = n. Then cd X (resp. q(X)) ? max (1 + m, n) .

sequence,fora coherentsheafF PROOF. We apply the local cohomology on X. ***

>Hy~(XqF)

Hi(X9 F)

->Hi(X

- Y F)

>**

Thus we need onlyshow that Hj(X, F) = O

for i > l +

But this groupis the abutmentof a spectralsequencewithinitialterm 1'(F)) . Now this is zero for q > m by the second hypothesis. And for q ? m, we have [Lc 2.8] E2Pq

=

,

ROBIN HARTSHORNE

410

iJCq(F)= dir limr,&aqq (Or,, F),

whichis a directlimitof coherentsheaves on the schemes Y,, all of which 0 for p > 1. It followsthat have Y as theirreducedscheme. Hence Ep' the abutmentof the spectralsequence is zero forq + p > l + m. is a generalizationofa techniqueused Example. The previousproposition by Budach [1, ? 7.5] to show that the cohomologicaldimensionof P3 minus a non-singularrationalquarticcurve is one. The curve C lies on a quadric family,so we can findone surface. Indeed, the quadricsforma 9-dimensional of them, Q, containing9 points of the curve. Then Q containsthe whole curve,and Q is necessarilynon-singular.We take X= P3 - C and Y = Q - C in the proposition.Then Y is affine(as one sees easily),so 1 = 0. Y is locally definedby a single equation, so m = 1. And X - Y= P3 - Q is affine,so n = O.

In fact,thissame techniquewould workforany curve C in P3 whichlies on surface Q, such that Q - C is affine.We do not knowwhethersuch a surfaceQ exists forany irreduciblecurve C. However, we will provelater (7.5) by anothermethod,that the cohomologicaldimensionof P3 minusany connectedcurve is one. PROPOSITION1.5. Let X be a quasi-projective scheme, and suppose, for a particular integer i, either (a) or (b) below holds. (a) Hi(X, F) = 0 for all coherent sheaves F on X (b) Hi(X, F) is finite-dimensional for all coherentsheaves F on X, and Hi(X, F(n)) = 0 for n large enough (depending on F). Then the same condition holds for all j > i. In particular, (a) implies cd X < i, and (b) implies q(X) < i. PROOF. By inductionon j, it is enoughto treatthecase j = i + 1. Also,

bynoetherianinductionon the supportof F, we mayassumethatSuppF= X, that X is reducedand irreducible,and that the statementis provenfor all sheaves withsmallersupport. Futhermore,we mayassumethat F is torsion free,because thereis an exact sequence 0

-

t(F)

-

F-*

F/t(F) -

0

wheret(F) is the torsionsubsheafof F. Here t(F) has supportless than X. Let X be a projectivecompletionof X, and let F C X be a hypersurface sectionof X, whichcontainsX - X. Let Y = Y n X. Then Y is an ample divisor on X, say I = (9x(-d) for some integer d > 0, and X - Y = X - Y sheaf is affine.Let j: X - Y >X be the inclusion,and let F be a torsion-free local of sheaf cohomologygives on X. Then the exact sequence

DIMENSION

COHOMOLOGICAL

0-

F

OF ALGEBRAIC

VARIETIES

411

X7C'(F)-+0.

, j*j*F

so Now j is an affinemorphism, 0

j*F)

Hq(Xj

forall q > 0. Hence (1)

Hi+'(X, F)

Hi(X, XC(F))

(We assume i > 1, since the statementis trivialfori = 0). We expressthe local cohomologysheaf as a directlimitof Ext's. (2)

F) = dir limk &&Tj1(C)x/Ik, the exact sequence Furthermore,

X 1(F)

whereI =

-y =

Ox(-d).

0

)

9X/jk+l

jk/jk+1

,

,X/Ik

0

gives rise to an exact sequence

3

0 -*

j-1 (eX /Ik, F)

> &xTj1 (OX/Ik+l,

(3)T1

(Ik/Jk+l,

F) F)

-

0 .

and theEx Note thatthe &cjr0= Cemare zerobecauseF is torsion-free, forq > 1 because Y is a divisor. We also note that Ik/I k+l

=

Hence (4)

&TJ1

(Ik/Jk+l

(I/I2)k

=

(y(-kd)

F) --xrl

((C), F)(kd).

We now returnto our problem,which is to caulculate Hi+'(X, F). Assuminghypothesis(a) or (b), we mustshow that this groupis eitherzero, or and zero forF(n), n large. finite-dimensional, Suppose firsthypothesis(a). Using (2), we see that XIQ(F) is a direct limitof coherentsheaves on X. Hence Hi of it is zero, which by (1) gives the result. Suppose on the otherhand we have hypothesis(b). Then applyingour hypothesisto the sheaf G = &cfS1(0y,F),

for all m, and zero for large m. we have Hi(X, G(m)) is finite-dimensional Hence by (4) and (3) we see that the directlimit H (X, X' (F)) = dirlimkH (X, &Tjr' (e9X/IkF)) so the limitis is eventuallyconstant,and all its termsare finite-dimensional, F is the is what we twisting which want. Finally, by (1) finite-dimensional, same as twistingG, so by twistingenough, we can make all the termsin that directlimitzero,and so Hi+'(X, F(n)) = 0 forn large.

ROBIN HARTSHORNE

412

2. The local analogue In this section we exploit the analogy between propertiesof varieties and properties of rings. We will carry over the notion of cohomological dimension and the integer q to a pair consisting of a ring A and an ideal J. We will prove in the next section a local version of Lichtenbaum's theorem, using techniques of commutativealgebra, which then allows us to recover the original global Lichtenbaum theorem. All rings will be commutative and noetherian, of finiteKrull dimension. To explain the analogy between projective varieties and rings, we recall the following well-known result [LC, EGA]. Let X be a projective variety, with homogeneous coodinate ring A. Let Y c X be a closed subvariety whose homogeneous ideal is J c A. Let M be a graded A-module, and let F = M be -the associated quasi-coherent sheaf on X. Then the cohomology of F on X - Y and the local cohomology of M, with supports in J, are related by an exact sequence of four terms,

O-

Hj(M)

M-

Z H(X-

>

Y. F(v))-,

HJ(M) -

0

and isomorphisms Z

Hi(X

-

Y F(v))

HJ+'(M)

for i > 1. We definethe cohomological dimension of a ring A with respect to an ideal J, written cd (A, J), as the least integer n such that HJ(M) = 0 for all i > n and all A-modules M. Thus for example if A is a local ring of dimension n, with maximal ideal m, then cd (A, m) = n [LC ? 6]. The case when J is the maximal ideal of a local ring should be thought of as corresponding to the case of a complete variety. On the other hand, cd (A, J) = 1 if and only if Spec A - V(J) is an affinescheme, as one sees easily by examining the exact sequence of local cohomology on Spec A. We have cd (A, J) = 0 if and only if V(J) is empty, i.e., J is contained in the radical of the zero ideal, or equivalently, J is nilpotent. To generalize the integer q, we note immediatelythat the local cohomology modules are very rarely finite-dimensional. In the analogy with projective varieties above, they correspond to direct sums of infinitelymany finitedimensional vector spaces. However, they are often cofinite,in the following sense. Let A be a local ring with maximal ideal m. An A-module M is cofinite if M has support at the closed point V(m), and if HomA (A/m,M) is a finitedimensional vector space over k = A/m. (See [LC ? 4] for some equivalent conditions.) The cofinite modules form an abelian category, stable under

COHOMOLOGICAL DIMENSION OF ALGEBRAIC VARIETIES

413

takingsubmodules,quotientmodules,and extensions. Now let (A, m) be a local ring, and let J be any ideal of A. We define q(A, J) as theleast integern > -1 such thatthe modulesHJ(M) are cofinite forall i > n and all A-modulesM of finitetype. (Wheneverwe writeq(A, J) it will be understoodthat A is a local ring.) Then fora local ringA of dimension n, and an ideal J, we always have -1 < q(AJ) < cd(A,J) < n . If J =m is the maximalideal, thenit is known[LC] that all the local cohomologymodulesHm(M)are cofinite,so q(A, m) -1. Note again theanalogy withcompletevarieties. Let us observea few formalpropertiesof cd and q. Clearlytheydepend onlyon the radical of J, and not upon the ideal itself. Thus we can always ofprimeideals,if we wish. Comparinga ring supposethatJ is an intersection A to the reduced ring Ared,and letting Jred= J-Are we see immediately that

cd (Ared,

Jred)

= cd (A, J) =

q (A, J) If Pi, *, pr are the minimalprimeideals of (0) in A, corresponding to the irreduciblecomponentsof A, we let Ai = A/pi,and Ji = (J + p)/Pi. Then q

(Ared, Jred)

cd (A, J) = max (cd (Ai, JJ))

q(A, J) = max (q(Ai, J)) . Mimickingthe proofof (1.1) we obtainthe following A' be a homomorphism of rings such that A' is a finite A-module. Let J be an ideal of A, and let J' = JA'. Then PROPOSITION 2.1. Let f:

A

-

cd (A', J') < cd (A, J) and q(A', J') < q(A, J) and there is equality in both cases if f is injective.

ofA, whenever This propositionapplies in particularto thenormalization one knowsthat it is a finitemodule. to considerjust In calculatingcd and q, we shouldnotethatit is sufficient modulesof finitetype(forcd), or just locallyfreemodules,or in factjust the singlemoduleM = A itself. This is because local cohomologycommuteswith direct limits,and is always zero fordegreesbiggerthan the dimensionof A [LC, 1.12].

Finally, we have the followingstatementcomparinga local ringto its completion. PROPOSITION2.2. Let A be a local ring, J an ideal in A, A the completion of A (with respect to its maximal ideal m), and let J = JA. Then

414

ROBIN HARTSHORNE

cd (A, J) = cd (A, J) q(A, J) = q(A, J). to studyHj3(M) forM=A. PROOF. As we remarkedabove, it is sufficient

We have

Hji(A) = dirlimpExti (A/l", A) Now the A/J"are modulesof finitetype, so the Ext's are compatiblewith tensoringby A, as is also the directlimit. Thus we see that HJ(A) = Hj1(A)0A A.

flatoverA, Hj(A) is zeroifand onlyif Hj (A) is zero Now since A is faithfully one has supportat the closedpointif so cd (A, J) = cd (A, J). Furthermore, and onlyif the otherdoes, and finally HomA (A/m,Hi(A)) =

Hom- (A/m', HJ(A)),

since the one on the left is killed by m, and so is not affectedby tensoring withA. Hence q(A, J) = q(A, J). Remark. The sameproofwouldworkforany local ringwhichis a faithfullyflatextensionof A. Now we will mentiona few examplesof these notions. Example 1. If A is a local ring of dimensionn, then HI(M) = 0 for i > n and all A-modulesM [LC, 1.12]; all the modulesH4(M) are cofinite if M is of finitetype, and in particular,Hm(A)# 0 [LC,6.4]. The results referredto were stated onlyforlocal ringswhichwerequotientsofa regular local rings,but we can reduceto thiscase byconsidering A. Thus cd(Am) =n and q(A, m) = -1. Example 2. Let A be a noetherianring,and let J be an ideal generated by r elementsx1,... , x,. Then cd (A, J) ? r. Indeed, for i > 2, HJi(M) Hi-'(X- Y F), whereX = Spec A, Y = V(J), and F = M. But X-Y is a unionof the r affinesX,, so H1(X-Y, F) = 0 forj _ r. If theelementsx,,***,xrforman A-sequence,and J#A, thencd(A,J)= r. Indeed we have HJi(M)= dirlimk Exti (A/J(k), M) foranyi, M whereJ(k) = (x4, - *-, x4). This holdsbecause thesystemofideals J(k)is cofinalwiththe powersJkof J. On the other hand, x4,... , x1 is also an A-sequence,so A/J'k)has a freeresolutionoflengthr given by the Koszul complex. Thus the Exti all vanishfori > r. For i = r and M = A, we get an isomorphism Extr(A/ljk),A)

A/eJ(k)

415

COHOMOLOGICALDIMENSION OF ALGEBRAIC VARIETIES

using the Koszul resolution. Underthis isomorphism the maps of the direct systembecome tqk AIJ'k'

Alj(k+l)

wherePk is multiplication by x1 * xir. From the propertiesof A-sequences it followsthat thesemaps are all injective[11], hencein particular A/Jc Hj(A) and so Hr(A) # 0. Thereforecd (A, J) = r. If furthermore A is a local ringof dimensionn, and r 1. This examplecorrespondsto the case of a set-theoreticcompleteintersectionin projectivespace. Example 3. Let A be a regularlocal ring of dimension4, with regular

parameters x1, x2,

X3,

x4.

Let

J

=

(xI 9x2)

n (x3,x,).

(This correspondsto two skew lines in projective3-space.) Then q(A, J) = 2 and cd (A, J) = 3 . As we remarkedearlier,it is sufficient to calculate HJ(A). We do thisby writingJ = J1f J2,J1= (x1,x2),J2= (x3,x4), and using the Mayer-Vietoris sequence forlocal cohomology, . . .

-

HJ1+J2(M)

HJ1(M) e HJ2(M)

-

Hiini2M)

-J ) HJnJ+2(M)

.**

We find,using the resultsof the previousexampleforJ,and J2,that H (A)

=

HJ1(A)& HJ2(A)

Hi(A) = Hm4(A)

HJ(A) = 0

fori > 4 .

Since H4(A) is cofinite,and HJ1(A)and HJ2(A)are not cofinite,we get

q(A, J) = 2 and cd (A, J) = 3.

PROPOSITION 2.3. Let A be a ring, and J an ideal.

Assume that for a

given integer i, either (a) or (b) below holds. (a) HJ(M) = 0 for all A-modules M. (b) A is a local ring, and HJ(M) is cofinite, for all A-modules M of finite type. Then the corresponding condition holds also for all j > i. In particular,

416

ROBIN HARTSHORNE

(a) implies ed (A, J) < i, and (b) implies q(A, J) < i.

to treat the case j = i + 1. PROOF. By inductionon j, it is sufficient to consider Also, by noetherianinductionon the supportof M, it is sufficient the case whereA is a domain,and M is a torsion-free module. Assumingcondition(a), suppose that Hj+1(M) # 0, and let x GHJ+'(M) be a non-zeroelement. Then x has supportin V(J), so is annihilatedby some non-zeroelementa e A. (The case V(J) = 0 is trivial.) We considertheexact sequence MaM

aM

yM

>O.

whence ...

-

HJ(M/aM)->

a+- Hh(M)

H+'(M)

-

...

This shows that HJ(M/aM) $ 0, which is impossible. So we concludethat HJ+'(M) = 0, as required. Now assumethatcondition(b) holds. The sameargumentas before,taking x e Hj+'(M) to be a hypotheticalelementwhosesupportis notin V(m), shows that indeedHj+'(M) does have supportin V(m). To show that it is cofinite, we must show in additionthat HomA (k, Hj+'(M)) is finite-dimensional.The are the elementsof Hj'+(M) killedby m. In images of these homomorphisms particular,theyare killedby any non-zeroelementa e m. Thus HomA (k,

Hji'(M))

=

HomA

(k, R)

whereR is the kernelof multiplication by a in Hj+'(M),

o

>R

, Hj+'(M)

a-%

Hi+'(M).

But R is a quotient of HJ(M/aM), whichis cofinitebyhypothesis,so R is also cofinite,and HomA (k, R) is finitedimensional. Note the analogyof thispropositionwith (1.5). The proofis simplerin this case, partlybecause condition(b) here is strongerthan the analogue of condition(b) there. 3. The local vanishing theorem This is the local analogue of Lichtenbaum's theorem. THEOREM3.1. Let A be a noetherian ring, J an ideal, and n an integer. Assume for every maximal ideal m containing J, that either

(a) dimAmn < , or

(b) dim Am = n, and V(J) meets every formal branch of Spec A at m in at least a curve; in terms of ideals, this means that for every minimal prime ideal p* of (0) in Am,

417

COHOMOLOGICALDIMENSION OF ALGEBRAIC VARIETIES

dim Am/(JAm + p*) > 1. Then HJ-(M) = 0 for all A-modules M. PROOF. Since Hj(M) has supportin V(J), it is enough to show that its

localizationat each Amis zero, form a maximalideal containingJ. Thus we reduceto the case of a local ring. If dimAm< n, the resultis alreadyknown. So we may assume that A is a local ringof dimensionn, withmaximalideal m. Futhermore,usingthe techniquesof theprevioussection,we mayassume that A is complete,irreducible,reduced,and normal. (The integralclosure of a completelocal domainis a finitemodule[19,(32.1)].) Then our hypothesis on J says simplydimA/J> 1. Let p D J be a primeideal suchthatdimA/p= 1. Then we claimthat for any A-moduleM, H;"(M)

,Hj"(M)

is surjective. (We may assume n > 2 throughout,because the theoremis trivialforn = 1, and vacuous forn < 0.) Indeed,let X = Spec A. Then

H;(M) Hj(M)

H j-1(X - V(t) Hn-1(X

-

Mi2)

V(J), M).

Applyingthe local cohomologyexact sequence to X subset V(J) Hn-1(X

-

-

-

V(t) and the closed

V(t), we have

> H-

V(Q), AM)

(X-V(J),

M)

> Hv-(J)_v(p)(X -VQ),

M).

This last module is zero, because X - V(t) is a schemeof Krull dimension < n [LC 1.12]. Thus we can replace J by p, and we are reducedto provingthe following statement. If A is a complete,normal,local domainof dimensionn, and p is a primeideal with dimA/p= 1, then H;(M)

= 0

forall A-modulesM. Considerthe topologyon A formedby the symbolicpowersp(k) of P. We claimthe p'k)_topology is equal to thepk-topology,formedbytheusual powersof p. Indeed,each pk C p(k). On theotherhand,byChevalley'stheorem[26,III ? 5 Th. 13], the m-topology is minimal,since A is complete. The intersectionof

n

p(k) is zero, because A is a domain. Therefore each the symbolic powers power of m contains some ps). Now since dim A/p = 1, the primary decomposition of pk is

ROBIN HARTSHORNE

418

] p(k)

pk=

nq

,

whereqkis primaryform. Hence qkcontainsa power of m, which contains a symbolic power p's, so pk D

p(max(k,s))

Hence the two topologies are equal. (This result is a special case of [26, VIII ? 5, Cor. 5].) Now writeA as a quotientof a completeregular local ring B. This is alwayspossiblesince A is complete. Let dimB = r. We considerthespectral sequenceof changeof rings E2

Em = ExtB (N, B)

= Extv (N, ExtB (A, B))

for any A-moduleN. Taking N and setting

=

A/pk,

passing to the directlimitover k,

Ext' (A, B)

=i

this spectralsequencegives in the limita new spectralsequence = H,,(f7q)

Em = dir limk Ext,

(A/pk,

B) .

we can replaceA/pk Since the pk-topologyon A is equal to the p'k)_topology, of using symbolicpowers The virtue the on right. in the expression by A/!(k) local ringA/p(k) has no embeddedprimesof zero, is that the one-dimensional and hence has depth 1. Therefore,as a B-module,A/4p(k)has homological dimensionr - 1, because B is a regularlocal ring[23, IV Prop. 21]. Thus Extr (A/!(k),

B) = 0

for each k, and so the Er termof the abutmentof the spectral sequence above is zero. We deduce that for p + q = r, E~q = O and in particular En

r-n

=

0

We will use this fact to deduce that Hn(f2r-n)

= 0

afterthe followinglemma. A be a homomorphism, where B is a regular LEMMA3.2. Let f: B local ring of dimension r, A is a normal local domain of dimension n, and define fi = Ext' (A, B). Then

419

COHOMOLOGICALDIMENSION OF ALGEBRAIC VARIETIES

(1)

r-

SY=0fori 1. PROOF. (1) Since B is a regular local ring, it has depth r, and I-depth n, where I is the kernel of f, since dim B/I = n. [LC, 3.6] Hence Exti (N, B) = 0

for any N with support in V(I), and i < r - n. (2) If e E Spec A is the generic point (i.e. $ = (0)), then zr-n= Extr-n (At, Be)

But Be is a regular local ring of dimension r - n, with residue fieldA, =k($)q and so this Ext is isomorphic to k(e) itself [LC pp. 63, 64]. Thus flr-n has rank one at i. (3) For this statement, we must show that if j > 1, and $ E Spec A is a point of dimension > n - j - 2, then gr-n+ j= =0.

Let

$

have dimension n

j

-

-

1. Then

f2r-n+j=

(At, Be),

Extrn+j

where Be is a regular local ring of dimension r - n + j + 1, and A, is a normal local ring of dimensionj + 1. Thus A, has dimension > 2, and being normal, has depth > 2. Therefore A, as a B,-module has homological dimension < r - n + j-1. Thus the Ext in question vanishes, and

a r-n+j

- 0

Proof of theorem, continued. We return to the spectral sequence discussed above. We know that Enr-n

=

0

and we wish to show that En r-ne - H;P(%r-n)

is zero. For this we must investigate those d's of the spectral sequence which involve this term. Outgoing d's land in subquotients of En+i+l'r-n-i = Hn+1l2r-noj)

= 0

by the lemma, part 1, or by the fact that dim A fromsubquotients of E2n-j-lr-n+i

=

HU-i-l(&2r-n+)

by the lemma, part 3. Hence we deduce that

=

0

j > 1 n. Incoming d's come j

>

1

ROBIN HARTSHORNE

420

H;(

-

r-,n)

0.

But now we note that H;$(M) = 0 for any A-module with support of dimension < n. By the Lemma, part 2, (rr-nhas rank one at the generic point of A. Hence also HomA (2r-n, A) has rank one at the generic point of A, so we can finda homomorphism >AA

&2 f: fQr-,n

-

which is an isomorphismat the generic point. Then we have Im f

0

Coker f

>A

--> .

But H; (Imf) = 0

because Im f is a quotientof ar-n; H; (Cokerf) = 0 because Coker f has support of dimension < n. Hence 0

H;(A) and so =

H;(M)

0

for all A-modules M. q.e.d. COROLLARY3.2 (Lichtenbaum's Theorem). Let X be a quasi-projective variety of dimension n. Then the following conditions are equivalent. (i) All irreducible components of X of dimension n are non-proper. (ii) HI(X, F) = 0 for all coherentsheavesF on X. PROOF. Assume (i). Let X be a projective closure of X. Let A be the homogeneous coodinate ring of X, and let J be the homogeneous ideal of the closed subset X - X. Then the pair (A, J) satisfies the hypotheses of the theorem for n + 1, so Hj; "(M) = 0 for all A-modules M. But every coherent sheaf F on X is of the formM, for a suitable graded A-module M of finite type, so using the remark at the beginning of ? 2, we have E

In particular,HN(X, F)

e7

=

Hn(X, F(2)))

=

Hj"+;(M) = 0 .

0.

Conversely, suppose some componentof X of dimensionn is proper. Then we need only find a coherent sheaf F on that component for which H1(X, F) # 0. So we may assume X is irreducible and proper. Then X =X,

COHOMOLOGICALDIMENSION OF ALGEBRAIC VARIETIES

421

J = m, the irrelevant ideal. It is sufficientto show that Hm+I(A)

0,

by the same remark as above. To do this we may localize at m, in which case we have a local ring of dimension n + 1, for which the result is known [LC, 6.4]. Note that the firstproofof this theorem,by Grothendieck,used a delicate argument about compatibility of local and global duality [LC, 6.9] and [RD, VII 3.5]. The second proof, by Kleiman [161, was valid more generally without the hypothesis that X is quasi-projective. COROLLARY3.3 (Theorem of Nagata [18, Th. 5]). Let X be a normal affinesurface over a field k, and let Y be a closed subset of pure codimension is affine. one in X. Then X-Y PROOF. In general, if X is an affinescheme, and Y is a closed subset, we consider the exact sequence of local cohomology, for any coherent sheaf F on X *F

>H'(X,

F) --)H'(X > Hy2(X9 F)

- Y F) ) H'(X,

F)

>*

.

The two outside groups are zero, since X is affine. Hence, using Serre's criterion,X - Y is affineif and only if Hy(X, F) = 0. Again, since X is affine, Hy2(X,F)

=

HO(X, JC2(F)),

so the conditionis equivalent to saying JC2(F) = 0, which is purely local along Y.

In our case, we apply (3.1) to each local ring A of X at a point y E Y. Since A is normal, it is analytically irreducible [19, (37.5)], so there is only one analytic branch, and the hypotheses are satisfied. Hence X'C (F) = 0 for all quasi-coherent sheaves F, and we are done. Example 1. If X is an affinescheme, all of whose local rings are UFD (for example a smooth scheme), and if Y is a -closed subset of pure codimension one, then X - Y is affine. For in that case, Y is locally defined by a single equation, and so 2Cy(F) = 0 for all coherent sheaves F (see ? 2, Example 2). Example 2. (Nagata [18]). If X is a normal affinevariety of dimension > 2 over a field k, and Y is a closed subset of X of pure codimension one, it may happen that X - Y is not affine. For example, let Q be a non-singular quadric surface in P3, let L c Q be a single ruling of one family, let X be the affinecone over Q, and let Y be the cone over L. Then X is normal of dimension 3, in fact it is a complete intersection, and hence also Gorenstein. Y is

422

ROBIN HARTSHORNE

of pure codimensionone, but X - Y is not affine. Indeed, as above, it is sufficient to finda coherentsheaf G on X such that HI(X, G) # 0. Let G correspondto a moduleM overthe affineringA of X, and let F = M be the correspondingcoherentsheaf on Q = Proj A. Then we saw at the beginningof ? 2 that Hy(X, G) =

-

L, F(v))

But Q - L is not affine(it containssome completecurves, namelythe other rulingsin the same familyas L), so we can findF, and hence G, to make these groupsnon-zero. 4. Cohomologyof formalschemes In this section,we make a few generalremarksabout the cohomology of coherentsheaves on formalschemes. Then,if X is a properschemeover k, Y a closed subscheme,and X the formalcompletionof X along Y, we relate questionsabout the cohomologyof X - Y to questionsabout the cohomology of X. The notionof formalschemeis definedin [EGA, I. 10]. All our formal schemeswill be noetherian. If I is an ideal of definitionof a formalscheme 'C, thenthe closed subschemeof 'C definedby I, say X, will be called a subschemeof definitionof ?X. Thereis a smallestsubschemeofdefinition, which we call thereducedsubschemeof definition.If X is a subschemeof definition to an ideal of definition corresponding I, then we will denote by X_,n > 0O definedby I+', so that X =XA. And if i is a the subschemeof definition coherentsheaf on 'CX,we will denoteby Fn the sheaf Y 0 (Dxnon Xn. If one subschemeof definition is affine, so are all the others[EGA, I, 5.1.9], and we will say that 'C is affine. If fX is a formalschemeover a fieldk, and of OCis properoverk, so are all theothers[EGA, if one subschemeof definition II. 5.4.6], and we will say that WXis properover k. (Note the deviationfrom the terminologyof [EGA].) If OC admits an invertiblesheaf 09c(1) whose restrictionto some subschemeof definitionis ample, then its restrictionto is ample [EGA, II. 5.4.14], and we will say that everysubschemeof definition EC is quasi-projectiveover k. If OCis quasi-projectiveand proper,we say it is projectiveover k. A formalschemeEC is regular if all of its local ringsare regularlocal rings. For example, the formalcompletionof a non-singularvarietyalong an arbitraryclosedsubset is a regularformalscheme. From now on, all our formalschemeswill be regular. If OCis a (regular) formalscheme,and X is we say thatX is locally a completeintersection,if a subschemeof definition,

COHOMOLOGICALDIMENSION OF ALGEBRAIC VARIETIES

423

its sheaf of ideals I is locallygeneratedby an Con-sequence.It followsthat X is a Gorensteinscheme,that I/I2 is locallyfreeon X, and that IT/In+1S"(I/I2), the nthsymmetricpower. The dual of I/I2, namely Homx (1/12,Ox) is called the normal bundleof X in EC. PROPOSITION4.1. Let JCbe a formal scheme, proper over k, and let iFbe a coherent sheaf on EC. Then for each i > 0O

Hi(EC, iF)= invlimaHi(Xn, FI) . PROOF. This is a straightforwardapplication of [EGA, 0III.

13.3.1]. We

note firstthat [EGA, I. 10.11.3].

iY= inv lim Fn

We take as base forthe topologyon X0 the affinesubsets U,. These are also affinesubsetsof X", foreach n. Hence the projectivesystem(Hi(UA,FJ))"'? is surjectivefori = 0, and identicallyzero for i > 0, so in eithercase satiscondition(ML). The systemof sheaves (F.) is surjecfiesthe Mittag-Leffler tive. And since each X. is properover k, the cohomologygroupsHi(X,, FJ) so that projective systems (H'(Xn, Fn))n?0also are all finite-dimensional, satisfy (ML). Under these conditions,the result quoted above applies, and show that Hi(OX,Wi)

nv limeHi(X", Fn).

be a direct system of k-vector spaces, let W be LEMMA4.2. Let (Vq)q,0 another k-vectorspace, and let A: dir lim V,,

W

be a morphism. Then p is injective (resp. surjective, resp. bijective) if and only if the map j

W'

inv lim Vn,

'where' denotes dual vector space, is surjective (resp. injective, resp. bijective). of dirlimand invlim,we have PROOF. By definition

(dirlimV,)' = invlimV' . So the resultfollowsfromthe fact that a map of vector spaces is injective, surjective,or bijective,if and onlyif thedual map is surjective,infective,or bijective. Note thatthevectorspaces involvedneed notbe finite-dimensional. PROPOSITION4.3. Let X be a non-singular proper scheme over k of dimension n, let Y c X be a closed subset, and let X be the formal completion of X along Y. If F is a coherent sheaf on X, we denote by

424

ROBIN HARTSHORNE

F =F(F)o/

x

its formal completion along Y. Let i be an integer. Let c) = M2, be the sheaf of n-differentialforms on X over k. Then (a) Hn-'(XY F) is finite-dimensional for a locally free sheaf F on X, if and only if Hi-1(X, G) is finite-dimensional for G = F (0 co.

(b) Hn-i(X

Y, F) = 0 for a locallyfree sheaf F on X, if and only if

-

ari-1:Hi-1(X, G)

Hi-'(Xg G)

is surjective, and aji: H'(X, G)

>Hi(X9 G)

Co. is injective, for G = F PROOF. We use the long exact sequenceof local cohomology

-

H-(X,

Hyn-i(X, F)

) Hyn-i+'(XgF)

F)

-

h-t~ )HH-(X9

H-

(X-Y

F)

F)

*

Since X is proper,the groups in the middlecolumnare finite-dimensional. if and onlyif Hyn-i+'(X, Thus Hn-i(X - Y, F) is finite-dimensional F) is finitedimensional.But Hy"-i+1(X,F) = dir limk Ext ni+l(Cx/Ik F)

whereI = Iy is the sheaf of ideals of Y. This Ext can be rewrittenas C), Extf-i+1(cx/Ik 09F 0(, whereF = Hom(F, Qx)is the dual sheaf. Let G = F(& a. ThenusingSerre dualityon X [RD,VII ? 4], this Ext groupis dual to Hi-'(X,

Ox/Ik

0

G) .

intoan inversesystemof cohomolThe directsystemof Ext's is transformed ogy groups. So using the previous propositionand lemma, we findthat if and only if H'-'(X, G) is finite-dimenHyn-h(X,F) is finite-dimensional sional. This provespart (a) of the proposition. For part(b), we note that Hn-i(X - Y, F) = 0 if and onlyif z8, is surjective,and ,8n, is injective. Then,sinceH"-i(XF) is dual to H'(XF(0 co), we applythe same reasoningas above, and the lemma,to get our result. COROLLARY4.4. In the proposition, part (a) or (b), the statement involving a sheaf F is true for all locally free sheaves F (resp. for all invertible sheaves F) if and only if the statement involving a sheaf G is true for all locally free sheaves G (resp. for all invertible sheaves G).

COHOMOLOGICALDIMENSION OF ALGEBRAIC VARIETIES

425

PROOF. This followsfromthe fact that as F runs over the set of all locallyfreesheaves (resp. all invertiblesheaves), so does

G = Fo(

.

Remark. This proposition givesa simpleproofof Lichtenbaum'stheorem forthe case of a non-singular varietyX minus a closed subset Y. We take i = 0. Then we have Hn(X - Y, F) = 0 forall locallyfreeF, and hence for all F, if and onlyif H0(X, G)

-

H0(X, G)

is infective,forall locallyfreeG. If Y is empty,thentakingG = ex shows this does not happen. However, if Y is non-empty,then since G is locally free,everysectionhas supporton all of X, and so it gives a non-zerosection of G over X. 5. The case of an ample normal bundle

The resultsoftheprevioussectionsuggestthatwe studyformalschemes, and ask whenthe cohomology groupsoflocallyfreesheavesare finite-dimensional in low degrees. In particular,lookingat HI, we can ask when is = k? This is the questionof holomorphic functions,raised by Zariski H?(090f) of can sheaf We also consider the total quotientrings of 0C9,and JCK [241. ask whetherthe fieldof meromorphic functionsK(fX) = H0(XJC)has finite transcendencedegreeover k. In the case of an amplenormalbundle,we can prove several resultsin this direction,generalizingthose of [AVB,? 8]. See of an amplevectorbundle. [AVB, ? 2] forthe definition THEOREM5.1. Let EXbe a regular formal scheme, proper over k. Assume that there is a subscheme of definition X, which is locally a complete intersection in 1DC,and such that the nomal bundle E = (I/2)V is ample on X. Then Hi('C, i) is finite-dimensional for all locally free sheaves Y on DC, and for (all i < d = dimX, l

=YO.

if chark = O

if d > 1, char k = p > O, and E is a quotient of a direct sum of ample line bundles.

PROOF. The proofis the same as the proofof [AVB,8.1], but we will repeat it, because the hypothesesare moregeneral. Let Y be a locallyfree

sheaf on EC. Let Xn,be the subscheme definedby I"+', and let F. = TY(0 C9x.

Then by (4.1),

Hi(EC,JF)= invlimHi(Xn,F).

ROBIN HARTSHORNE

426

The exact sequence 0 I"/-"+

c9X n+1

X

O-

0

gives a colomologysequence -

F0 Ir/I-+) -* Hi(X, Fo

Hi(Xn+l,Fn+?)-> Hi(Xn, Fn)

-

**

to show that Thus to proveour result,it will be sufficient Hi(X, F0 ?gIJ/Is+') = 0

foreach i and forn large enough. on X [RD,V 9.3]. Then by Let GObe the sheaf of dualizingdifferentials dualityon X, this cohomologygroupis dual to EXtd-i

(Fo (g

aInl())

Now In/I'+1 = S,(I/I2), and is locallyfree,so this Ext is equal to Hd-i(X,

F0

?

Sn(I/12)v

0

()

Letting G = F0 0 w, and writing = FP(E)

Sn(I/12)V

to show foreach coherentsheaf G on X, that it will be sufficient Hd-i(X, G 0&I'(E))

= 0

forn large enough. If chark = 0, thenF"(E) = S"(E), and since E is ample, this group is zero forall d - i > 0, i.e., forall i < d. If chark = p > 0, the resultfollowsfromthe lemmabelow. LEMMA5.2. Let X be a schemeof dimensiond, properoverk and let E be a quotientof a directsum of ample line bundleson X. Then Hd(X, G 0 F"(E)) = 0

for everycoherentsheaf G on X, and for n large enough(dependingon G). PROOF. Let E be a quotientof G),rlLi. Then J'"(E) is a quotientof pn($

Li) = Sn(eD Li).

But Hd(X, *) is a right exact functor,and tensoringwith S n(@i Li) makes cohomologyvanish [AVB2.2 and 3.3]. Remark. It is the absence of a suitable generalizationof this lemma which makes the statementof the theoremso awkwardin characteristicp. One mightconjecturethat if E is ampleon a properschemeX, then Hi(X, F

0

rP(E)) = 0

for all coherentF, all i > 0, and n large enough. Anothercloselyrelated

COHOMOLOGICALDIMENSION OF ALGEBRAIC VARIETIES

questionis whether

427

0 E(P-)) = 0

Hi(X, F

for large n and i > 0, i.e., whether E is "cohomologicallyp-ample"' (cf. [AVB ? 6]).

5.3. Withthesame hypothesesas the theorem,let 0O(1) be COROLLARY an invertiblesheaf on 'C, whose restrictiction(9(1) to X is ample. Then (for thesame i as in the theorem),

H'(EC, T(-m))

= 0

for m large enough. PROOF. By examiningthe proofof the theorem,we see that it will be to prove sufficient H'(X, F0(-m)) = 0 and Hi(X, Fo (0 In/In+'(-m)) = 0 forall n, and form large enough(independentof n). By duality,we mustshow that Hd-i(X, G(m)) = 0

and

Hd-i(X, G 0 rP(E)(m))

0

forall n, and form large enough(independentof n). In the case of char0, we considerthe vectorbundleE 0(,(l), also ample [AVB2.2]. Hence Hd-i(X, G 0 S7(E for all r > r,. But

0 CD(1)))=

whichis

0

SP(E)(q) Sr(E 0 (D(1)) = Ep+q=r p,q0O

In particular,if we take m ? r., then Hd-i(X, G

0) S-(E)(m)) = 0

forall n ? 0, and d - i > 0. In the case of charp > 0, we have E writtenas a quotientof a direct sum of ampleline bundles, 3)Li

E

,0.

We applythe same argumentto the bundleEDLi 0D(x(1), and concludethat Hd(X, G (0 S'n($ L,)(m)) = 0

forall n > 0, m

>

r,.

But this has

ROBIN HARTSHORNE

428

Hd(X, G 0&P-(E)(m)) as a quotient, so we are done. COROLLARY 5.4. With the hypotheses of the theorem, assume furthermore that 'X is connected and that d > 1. Then

H0(9, At) = k, i.e., there are no non-constant holomorphic functions on 'X. PROOF. (Hironaka). Under these hypotheses, H0(X, (ax) is an integral domain. However, it is also a finite-dimensionalk-vector space, and so must be equal to k, since k is algebraically closed (as always). Remark. For the completion of projective space along a subscheme, we will prove a stronger result below (7.3). 5.5. Let X be a non-singular proper scheme over k, of dimension n, and let Y c X be a closed subscheme of dimension d, which is locally a complete intersection, and such that the normal bundle E = (I/12)V to Y in X is ample. (We do not assume Y is connected.) Then Hi(X - Y, F) is finite-dimensional, for all coherent sheaves F on X - Y, and for COROLLARY

Iall i > i > n

if chark = O

n-d, -

1

if d > 1, char k =p, and E is a quotient of a direct sum of ample line bundles .

Furthermore, if (9x(1) is an ample invertible sheaf on X, then H'(X-Y,

F(m)) = O

for m > O

for the same values of i. PROOF. Since every coherent sheaf on X - Y extends to X, and since every coherent sheaf on X is a quotient of a locally free sheaf [17], we may assume that F is the restrictionof a locally free sheaf on X (see the discussion in ? 1). Then we apply (4.4) and the theorem, with =X= X, the formal completion of X along Y. Note that the statement about H'(X - Y, F(m)) vanishing for large m carries over, by (4.3) to the statement about Hi(X, G(- m)) vanishing for large m. LEMMA 5.6. Let Y be a non-singular subschemeof Pk. Then its normal bundle is ample. In fact, it is even a quotient of a direct sum of ample line bundles. PROOF. Let I be the sheaf of ideals of Y, and consider the exact sequence 1/12

d

&2pl0 (&

-

-_

0,

which obtains for any closed subscheme of P = P4. Here &2' represents the

COHOMOLOGICAL DIMENSION OF ALGEBRAIC VARIETIES

429

sheaf of relative differentialformsof P or Y over k. Since Y is non-singular, I/I2 and &?4are locally free sheaves on Y, and d is injective [RD, III 2.1]. Therefore E = (I/12)V is a quotient of Tp 0o Cy, where T denotes the tangent bundle. But Tp is itself a quotient of Dp(1)1f+1, so finallywe have a surjection >E

CY(1)-+1

>0

which shows that E is a quotient of a direct sum of ample line bundles, and hence is itself ample. COROLLARY5.7. Let Y be a non-singular closed subscheme of dimension d of P = Pk. (We do not assume Y is connected.) Then Hi(P - Y, F) is finite-dimensional, for all coherent sheaves F on P - Y, and for

{allion-d,

if chark = O if d ?1 and chark = p > O.

(i ? n -1, Furthermore, F(m)) =O,

Hi(P-Y,

for m > O,

for the same values of i. Remark. Grothendieck [SGA62 XIII. 1.3] conjectures that the same is true for any Y which is locally a complete intersection. Problem. In the context of (5.1) and its corollaries, it is natural to consider the graded ring

A=

H(QX:, @C(m)).

EmO

Each graded part is finite-dimensionalover k, so we can ask if A is a noetherian ring and if it is finitelygenerated over k. Also, if Syis a locally free sheaf on ~X, and i < d, we can consider the graded A-module Hi(U)

=

d

eIF(m))

Again, each graded part is finite-dimensional,and they are zero for m < 0, so we may ask whether Hi(U) is a finitelygenerated module. We will show in the next section that A is a normal integral domain, of transcendence degree < dim fX + 1 over k. Another problem. Find a local analogue of (5.7). This should say (conjecturally) that if A is a regular local ring of dimension n, and J c A is an ideal, such that V(J) is non-singularexcept for the point {m}, and dim V(J) = d, then HJ(M) is cofinite, for all A-modules M of finite type, and for all The hypothesis on J implies easily that these modules Hj(M) have i > n-d. support in V(m). However, it is not true (for arbitrary A, J, M, with M of

ROBIN HARTSHORNE

430

finitetype) that HJ(M) having supportin V(m) implies it is cofinite[13]. Hence the problem. 6. Meromophic functions on formal schemes

In this section we continuethe study of formalschemeshavinga subscheme of definitionwhich is locally a completeintersection,and whose normalbundleis ample. Ourmaintheoremstatesthatthefieldofmeromorphic functionson such a formalschemehas transcendencedegreeover the base fieldat mostequal to the dimensionofthe formalscheme. The originalversion of this theoremhas been muchimprovedby suggestionsof H. Hironaka. X, we will conIf ECis a formalscheme,witha subschemeof definition sideran invertiblesheaf CDc(1)on X, whose restrictionto X is ample. Our technique is to show that dim H0(X, @(.(v)) is bounded by a polynomial in ., function large. Then we will show that everymeromorphic forv sufficiently

on fX is a quotientof two sectionsof Ccjv), forsomev, and thus we will be able to estimatethe transcendencedegreeof K(QX).

LEMMA 6.1. Let X be an algebraic curve, let E be a vectorbundle of

rank t on X, let L be a line bundleon X, and letF be a coherentsheafon X. Let n > OveZ. h(n, v, F) = dim H'(X, F PFn(E) ( L>).

Then (a) If L is ample,thereis a polynomialPF e Q[z, w] oftotaldegreet,and thereare integerse > 0, NF > 0, (with e independentof F) such that for n > 0, for ) - ne < NF h(n, v, F) < /pF(fn, i) 0 for - ne > NF (b) If E is ample, thereare integersMF < 0, and f > 0, (with f independentof F) such that for v < MF, and n + fp > 0, we have h(n, 2, F) = 0 . PROOF. First we make a numberof reductions.If

0

> F'

>F

> F"

-

0

is an exact sequenceof coherentsheaves on X, and if the lemmais true for F' and F", thenit is true forF. Similarly,if it is trueforF, thenit is true forF". Thus we mayassumethatF is a sheafon someirreduciblecomponent of Xred. And since Ered,Lredare also ample [AVB ? 4], and the operationsF'I, 0Xv commutewith taking reduced subschemes,or irreduciblecomponents, we may assume that X is irreducibleand reduced. Futhermore,we mayas-

COHOMOLOGICAL

DIMENSION OF ALGEBRAIC

VARIETIES

431

sume that X is complete, for otherwise the cohomologygroups H' are all zero. If f: X' X is any finite, surjective morphism, then for any coherent sheaf F on X, there is a coherent sheaf G on X', and a generic surjection -

p: f, G

>F

(compare the proof of (1.1)). Hence for any locally free sheaf E on X, the map H'(X, (f*G) 0&E) H'(X, FO E) -

is surjective. Moreover,

(f*G) 0 E = f*(G Of*E) by the projection formula, and

H'(X, f*(G 0&f *E)) = H'(X', G 0&f *E) . Finally, E and L are ample on X if and only if f *E and f *L are ample on X' [AVB 4.3], and the operations P" and 0&v commute with f *. Thus it is sufficientto solve our problem on X', for f *E and f *L. In particular, we may assume X is non-singular,by passing to the normalization. Note that in characteristic 0, P"(E) = S"(E). In characteristic p > 0, we make a furtherreduction. Consider the scheme X, over k, which is the

same schemeX, but wherek acts by pth roots (cf. [AVB ? 6]). Then we have the Frobenius morphismf: X Xp, which is a finitek-morphism. We replace -

X by Xp, transporting the structure, then apply the discussion above to the finite morphismf. We have f *(Ep) = E(P), using the notation of [AVB ? 6]. Thus, repeating this process, we may replace E by E(pr) for any integer r > 0. In particular, since X is a curve, E is p-ample [AVB 7.3], so for r large enough, E (r, is a quotient of a direct sum of ample line bundles. (Take any ample line bundle 0(1). Then 0(-1) 0 E Pv' is generated by global sections for r > 0, so E'_r, is a quotient of a direct sum of copies of 0(1).) So we may assume E itself is a quotient of a direct sum of copies of an ample line bundle 0(1). If t is the rank of E, we can findt of these ample line bundles so that the map OM~t E is generically surjective. Then r"(l)-)

, P(E)

is generically surjective, so we can replace E by 0(l)t. In that case, p"(E) = S"(E). Thus we see that in characteristic p also, it is sufficientto prove the lemma for S"(E) instead of P"(E). So now we have a complete, non-singular curve X, a vector bundle E, a line bundle L. a coherent sheaf F, and we let h(n, v, F) = dim H'(X, F0(& S"(E) (0 L") .

432

ROBIN HARTSHORNE

We wish to provestatements(a) and (b) of the lemma. To prove statement(a) we assume L is ample, and write L = Cx(1). Using our firstremarkabove, we can makefurtherreductionson F. We may replaceF by a directsum of @(- vi), sinceany F is a quotientof such. Then we may considerone F = @(- vi) at a time. Finally,since in that case

F0

$ LP =

S'(E)

S"(E)

(0 Lv-vi

by makinga changeof variablein v, we may assume F = O, Since C(1) is ample,thereis an integere > 0 such that E(e) is generated by global sections(this definese). Then E is a quotientof a direct sum of copiesof @(- e). In particular,we can findt = rankE of themsuch that

E

(9(-e)t -

is genericallysurjective. ThentheHI groupsare surjective,so we can replace E by @(-e)t. Note now that n+ t-1E

Sn(E) 0$ LI = @(-ne)( t-) 0 @0( a

' 0(v(-ne)

'n+t -1E

(tl

)

.

Hence h(n, v) = h(n, P. Ox)

= (

dimH'(X, 0((

I 1)

-

ne)).

But we are workingon a non-singular curve,so thereis a polynomialp0e Q[z] of degree 1, and a constantN, such that dimH'(X, 0((

-

ne))

Ip,(v- ne)

(0

-

n

forv - e < 0 for -ne > 0 forv' - ne > N.

Indeed,we need onlytake p0to be minusthe Hilbertpolynomialof X. Now addinga suitableconstantto p0,to take care of the values 0 < v - ne < NY we findthereis a polynomialp, e Q[z] such that dimHX,

0(-ne)) ne,, N. v

Finally,we define p(n, v2)=

(

')P1(v - ne) ,

which is a polynomialof total degree t in n and v, and which fulfillsthe of part (a) of the lemma. requirements For part(b), we assumethatE is ample. Then L-' 0 SNo(E) is generated

COHOMOLOGICAL

DIMENSION

OF ALGEBRAIC

433

VARIETIES

by global sections, for no large enough, and hence L-1 0 S?o+'(E), which is a quotient of L-1' 0 Slo(E) (0 E, is ample [AVB 2.3]. Let f = no + 1, so that S (E) 0&L'1 is ample. Then E e Sf(E) 0 L-1 is also ample [AVB2.2], so there is an integer mo > 0 (depending on F) such that for all m > mi, H'(X, F 0 Sm(E e Sf(E) 0 L-1)) = 0 . Now this symmetricproduct is a direct sum, for p + q = m, and p, q > 0, of Sv(E)

0 Sq(Sf(E) 0 L-1),

and this has as a quotient Sp+qf(E) 0 L-q . Hence we have H'(X, F0(

SP+qf(E)

0 L-q)

= O

for p, q ! 0 and p + q > mi . - m. Let MF Then we claim that the conditionof part (b) is satisfied, namely that =

h(n, v, F) for < MF, and n + fr>O. Indeed, let q = nmo. and p + q = n + fr - v > 0 -

0 -

, p=nn + fr. Then p,q >O

THEOREM6.2. Let OC be a regular, proper formal scheme of dimension 4. Let X be a subscheme of definition, and assume ( i ) X is locally a complete intersection in DX, (ii) the normal bundle E = (I/I2)V is ample on X, 1. (iii) dim X Let Oc(1) be an invertible sheaf on EC, whose restriction @x(1) to X is ample. Let SYbe a locally free sheaf on 'X. Then there is a polynomial Pf e Q[z], of degree d, such that dim H(

7X2(u)) < PT(?)

v>

0

PROOF. First, we reduce to the case where dim X = 1. If dim X > 1, let Y c X be a hyperplane section (in some projective embedding; X is projec-tive,because it has an ample sheaf Ox(l) on it). Let OX'be the formal com-pletionof EC along Y. Then EC' is a regular formalscheme, with Y as a subscheme of definition. Y is locally a complete intersection in X'. Its normal bundle E' is an extension of E by the normal bundle N of Y in X, which is ample, since Y is a hyperplane section. Hence E' is ample [AVB 3.4]. Finally, note that

434

ROBIN HARTSHORNE

is injective (where F' = iF0& ). Indeed, since WY(v) is a locally free sheaf, any section has a non-zero stalk at any point y e Y. But the map of local rings , ey -*

X,

is injective, since the latter is the completion of the former with respect to a certain ideal. Thus it is sufficientto solve the problem for X2',Y, 1 '. By repeating this process a finitenumber of times, we reduce to the case dim X = 1. Let I be the sheaf of ideals of X, let Xn,be the subscheme definedby In, let F = = . Then O0 @8 Ox I9

H0($X,EF(v))= invlime H0(Xn,Fn(L)) by (4.1). Using the exact sequence 0

In/In+l

-

> eX

>(9y+1

> 0

we get

0

> HO(X,F(0)

0

IP/II~')

> H0(Xn+1,Fn+1(V))

>H?(Xnl Fn(v))) Hence

dimH0(TC,if(l)) < En o dimH0(X, F(v) 09I n/In+') If co is the sheaf of dualizing differentialson the complete intersection curve X, we can apply duality, and find

H0(X, F(v) (0 III/In+')I H1(X, F(1_) 0 (In/InIl+)v (0 ) But E = (I/I2)v, and In/Ini Sn(I/I2), so this latter group is just -

H'(X

IF 8& oi (& IPI(E)

9(;)

Now we can apply the lemma. Let p = p-,3,.. For v < MF, we have - ne < 0 < NF, SO h(n, v) < p(n, v),

and h(n, v) = 0 for n + fp > 0, i.e., n > -f v. Thus dim H0(XC, WIF()) d + 1, then one can find homogeneous elements yo, **, Yd+i in A, which are algebraically independent over k. By raising them to suitable powers, we may assume they are all of the same degree, e. Then A contains the polynomialring k[y0, ** , Yd,1], where each ys is of degree e, and so for any n > 0O dim Ane

(

d + 1),

which is a polynomialof degree d + 1 in n. This is impossible, so we conclude that tr d K(A)/lk< d + 1. (b) Let yo,* * , Yd be algebraically independenthomogeneous elements of A. We may again suppose they all have the same degree e. Suppose that K(A) is not a finitealgebraic extension of k(yo, * * *, Yd). Then one can findan infinitesequence of homogeneous elements z1, z2, * * * of A, which are linearly independent over k(yo, * * *, Yd). Considering the integers deg zi modulo e, we can findinfinitelymany with the same residue, say a. Thus, consideringonly those zi, we may assume that deg zi = a + efi,fi > 0, for all i. Now A contains the submodule L'

, , k[yO **

YJ *Zi

Hence for any n, dim Aa+en

(

> Efi