The development of Algebraic K-theory before 1980

Table of contents :
The pre-algebraic period......Page 1
The rise of topological K-theory (1959-1963)......Page 3
The structure of Projective Modules (1955-1962)......Page 4
The rise of algebraic K-theory (1957—1964)......Page 5
Applications to Topology (1963-1966)......Page 7
The Congruence Subgroup Problem......Page 8
Classical consolidation (1966-1968)......Page 9
Quadratic Forms and L-theory......Page 10
K_2 arrives (1967-1971)......Page 11
Higher K-theory arrives (1968-1972)......Page 13
Applications to Algebraic Geometry......Page 15
Homological methods......Page 16
The Lichtenbaum Conjectures......Page 17
Symbol Calculations......Page 18
Pseudo-isotopies and A( X )......Page 20
References......Page 22

Citation preview

The development of Algebraic If-theory before 1980 Charles A . Weibel Dedicated to H y m a n

Bass

Algebraic if-theory had its origins in many places, and it is a matter of taste to decide whether or not early papers were if-theory. We shall think of early results as pre-history, and only mention a few typical examples. One early example is the introduction i n 1845 of Grassmann varieties by Cayley and Grassmann. This formed the core of the classifying space notions in if-theory. Another example was the development (1870-1882) of the ideal class group of a Dedekind ring by Dedekind and Weber [DW], and its cousin the Picard group of an algebraic variety in the late 19th century [PS]. The Brauer group, introduced in the 1928 paper [Br28], and the W i t t group, introduced in the 1937 paper [Wi37], also belong to this pre-historical era. For various reasons, we will primarily limit ourselves to the period before 1980. In a few cases, we have followed tendrils which complete themes rooted firmly i n the 1970's. Like all papers, this one has a finiteness obstruction: only finitely many people and results can be included. We apologize in advance to the large number of people whose important contributions we have skipped over. The compendium Reviews in K - t h e o r y [Mag] was used frequently in preparing this article. Various other survey articles were very useful to us, including [Mi66, Sw70, Ba75, Lo76, Va76, 088]. We encourage interested readers to check those sources for more information, and textbooks like [Rosen] for technical information. The pre-algebraic period Algebraic K-theory had two beginnings, one i n geometric topology and one in algebraic geometry. The first was Whitehead's construction of the Whitehead group W h ( w ) of a group ж as an obstruction i n homotopy theory. The second, from which the subject got its name, was Grothendieck's construction of the class group K ( X ) of vector bundles on an algebraic variety X as a way to reformulate the Riemann-Roch theorem. The work of J . H . C . Whitehead on simple homotopy theory [Wh39] [Wh41] [Wh50] formed one beginning of algebraic if-theory. Suppose given a homotopy T h e author was supported i n part by N S F G r a n t s . ©1999 A m e r i c a n M a t h e m a t i c a l Society

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C H A R L E S A. W E I B E L

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equivalence / between two complexes (simplicial in 1939 and 1941, cellular in 1950). Whitehead found an obstruction, called the t o r s i o n r(/), for / to be built up out of "simple" moves (expansions and contractions). Unlike the Reidemeister torsion of /, which is a real number, Whitehead constructed r(/) as an element of a universal group, now called the Whitehead group W h ( w ) , which depends only upon the fundamental group ж. The thesis of Whitehead's student G . Higman [H40] contained some calculations of elements of W h ( n ) coming from units in ZTT. However, the group W h ( w ) proved to be intractable, and the subject was put aside until 1961. In 1961, Milnor used Reidemeister torsion in [MiOlj to disprove the Hauptvermutung, the conjecture that two polyhedral decompositions of a space always have a common subdivision. The Hauptvermutung dated back to Poincare's attempts (in 1895 and 1898) to prove that the Betti numbers ( i . e . , the homology groups) of a polyhedron are independent of the polyhedral decomposition used to compute them. If the Hauptvermutung were true, one could easily fix the geometric gap (found by Heegard) in Poincare's proof. J . W . Alexander had found a way around this gap in 1915, but the Hauptvermutung had remained open for over 50 years. In [MiOlj Milnor coined the term Whitehead group for the quotient G L ( R ) / E ( R ) of the general linear group G L ( R ) by the subgroup E ( R ) generated by the elemen­ tary matrices — the group we now call K \ ( R ) . Milnor also clarified that White­ head's torsion invariant lies in the quotient of this group by ±тг. This quotient would not be called the Whitehead group W h ( w ) until [Mi66], but let us get back to the story. The second origin of if-theory came from algebraic geometry, and was imme­ diately influential. It was the key ingredient in Grothendieck's 1957 reformulation of the Riemann-Roch theorem [BoSe] [Gr57 . Here Grothendieck introduced a group К ( A ) associated to a subcategory A of an abelian category. This was the real beginning of if-theory; Grothendieck chose the letter K for "Klassen" (the German word for classes); today we write K Q ( A ) for this group and call it the Grothendieck group of A . Grothendieck was interested in the abelian category M(.V) of coherent sheaves on an algebraic variety X , together with its subcategory P(.V) of locally free sheaves on X . He defined what we now call K ( X ) and G Q ( X ) as K P ( X ) and K M . ( X ) , respectively, and showed that these groups agree for nonsingular varieties. Then he showed that if Y is closed in X there is an exact localization sequence I

Q

G (Y)

G Q ( X )

Q

J

Q

G (X Q

-

Y )

Q

0.

Grothendieck also defined the 7-filtration on K Q ( X ) , and showed that the as­ sociated graded ring is isomorphic (up to torsion) to the Chow group A* ( X ) , via Chern classes c,: K Q ( X ) —f A ( X ) [Gr58 . Moreover, the total Chern character induces a ring isomorphism c h x '• K Q ( X ) ® Q = A * ( X ) ® Q. If /: Y —> X is proper he showed that the formula L

ет^£(-да/*(я] defines a covariant map / « : G Q ( Y ) —f G Q ( X ) , and used this to formulate the Grothendieck R i e m a n n - R o c h theorem: if X and Y are nonsingular and T x is the

THE

DEVELOPMENT OF ALGEBRAIC if-THEORY

3

Todd class of X then in A * ( X ) Q c h { f M ) • T x = f * ( c h ( y ) • Ту), x

у 6 K (Y).

Y

Q

The formula G Q ( X ) = G o ( X [ t ] ) was discovered by Serre in Fall 1957, and included in [BoSe, p. 116]. Here X [ t ] denotes the product of X with the affine line Spec(Z[i]). For nonsingular X this meant that K Q ( X ) = Ко(ХЩ). In particular, for X = Spec(fe) this implies that K o ( k [ x i , . . . , x ] ) — K o ( k ) = Z . In other words, every projective module P over R = k[x\, . . . , x ] satisfies P ф R = R for some m and n . n

m

n

n

The rise of topological Ж-theory (1959-1963) In 1959, Michael Atiyah and Friedrich Hirzebruch had the idea of mimicking Grothendieck's construction for topological vector bundles on a compact Hausdorff space X . In [AH59a], they observed that the tensor product of vector bundles makes the Grothendieck group K ( X ) into a commutative ring, and observed that Chern's original 1948 construction of Chern classes c, (combined into the Chern character ch) defines a ring homomorphism c h : K ( X ) —f H * ( X ; Q), much as in Grothendieck's setting. They defined relative groups K ( X , Y ) for closed subspaces Y, which are modules over K ( X ) . Using this module structure, they were able to prove a Riemann-Roch theorem for differentiable manifolds, also in [AH59a . A systematic study of this method led Atiyah and Hirzebruch to write their foundational paper [AH61], in which they defined the higher groups K ^ ( X , Y ) = K(S (X/Yj), n > 0. Observing that Bott's results on the periodicity of homotopy groups of the unitary group U could be expressed by an explicit "Bott period­ icity" isomorphism K ( X ) = K ( X ) , they extended K ( X , Y ) to all n using periodicity. Then they showed that the functors K ( X , Y ) form an extraordinary cohomology theory, satisfying all the Eilenberg-Steenrod axioms except the dimen­ sion axiom. They were able to calculate the if-theory of certain homogeneous spaces using what we now call the Atiyah-Hirzebruch spectral sequence: n

n

n

n + 2

n

n

м

Е

=

Я

Р ( Х , Ж * (point)) =>•

K

p + , 1

(X).

What made these new constructions so appealing was the power they brought to applications. We have already mentioned the Riemann-Roch theorem for dif­ ferentiable manifolds above [AH59a]. In [AH59b] they were able to show that complex projective space C P " cannot be embedded in for N = 4n — 2 a ( n ) , where a ( n ) is the number of terms in the dyadic expansion of n. In [AH62] they showed that if X is a complex manifold then any cohomology class и 6 H ( X ; Z) which is represented by a complex analytic subvariety must satisfy S q u = 0. This allowed them to disprove an overoptimistic conjecture of Hodge. 2 k

3

Ring homomorphisms t p : K ( X ) —f K ( X ) , which we now call Adams opera­ t i o n s , were constructed by J . Frank Adams in his paper [A62] (written in 1961). Using them, he was able to determine exactly how many linearly independent vector fields there are on S , which had been a famous outstanding problem. B y further analyzing the effect of the ф on the rings K ( X ) , Adams was able to describe the image of the Hopf-Whitehead J-homomorphism J : W i ( S O ) —f 7 r , ( 5 ) in terms k

n

к

n

n +

n

4

C H A R L E S A. W E I B E L

of the denominator of the Bernoulli numbers B ^ / i k . His ideas were based upon [AH59a], and discovered in 1961/1962, but his proof was not completely published until 1966 [А63]. At this point, the subject was off and running. B u t after 1962, the development of topological -fT-theory diverged somewhat from the development of algebraic A'theory. So let us put it aside and return to the algebraic side of things. The structure of Projective Modules (1955-1962) The notion of projective module was invented by Cartan and Eilenberg circal953, and first appeared on p. 6 of their 1956 book [CE], as a tool for working with derived functors. This set off to a search for nontrivial examples of projective modules, and an investigation into their structure. Some results were contained in [СЕ]. O n p. 11 we find the characterization of semisimple rings as rings for which every module is projective; later in the book (p. I l l ) , this was viewed as the characterization of rings of global dimension 0. For some rings, Cartan and Eilenberg could classify all (finitely generated) projective modules. If R is Dedekind (or more generally a P r i i f e r d o m a i n ) , it was pointed out (p. 14 and 134) that every projective module is a direct sum of ideals; this result was proven in Kaplansky's 1952 paper [K52], but in the language of torsionfree modules. This implied (p. 13) that all projective i?-modules are free when R is a principal ideal domain (or more generally a Bezout domain). Since the ring of integers in a number field is a Dedekind domain, this gave a classical source of non-free projective modules, discovered in the late 19th century. On p. 157 we find the statement that if R is a local ring then all finitely generated projective i?-modules are free. Kaplansky later showed [K58] that all projective modules over a local ring are free, as a consequence of the general re­ sult that any infinitely generated projective module is a direct sum of countably generated projective modules. For other rings, the classification was much harder. In Serre's classic 1955 paper [Se55, p.243], he stated that it was unknown whether or not every projective R module is free when R = k[x\, ...,XN] is a polynomial ring over a field. This became known as the "Serre problem." Serre's 1957 result that KQ ( R ) = Z meant that every projective i?-module P is stably free: P ф R = R for some m and n . Using this, Seshadri [Ssh58] immediately solved the problem for k [ x , y], but for n > 2 this proved to be a notoriously hard problem. It was not completely solved (affirmatively) until 1976, by Quillen [Q76] and Suslin [Su76]. M

N

Serre's 1955 paper [Se55] had a more long-reaching effect: it set up a dic­ tionary between projective modules and topological vector bundles. The analogy was strengthened by his 1958 paper [Se58], which showed that every projective i?-module has the form P ф R , where the rank of P is at most the dimension of the maximal ideal spectrum of R . This dictionary gave rise to a flurry of examples of non-free projective modules in the period 1958-1962. These came from algebraic geometry [BoSe, Se58], algebra [Ba61, Ba62], group rings [R59, Sw59, Sw60, R61] and topological vector bundles [Sw62]. Two later papers also fit into in this trend: [Fo69] and [Sw77] constructed many examples of projective modules based upon known examples of topological vector bundles. N

THE

DEVELOPMENT OF ALGEBRAIC if-THEORY

5

After 1962 people searched for projective i?[i]-modules which were not extended, i . e . , of the form РЩ. Schanuel's example [Ba62] of a rank one projective R [ t ] module that was not extended led to Horrock's criterion [H64], clarified in Murthy's papers [Mu65] [Mu66], and formed the basis of the notion of seminormality [Tr70]. Noticing that the rings involved were always singular, Bass conjectured in 1972 (problem I X of [Ba72]) that if R is a commutative regular ring then every projective i?[i]-module is extended; Bass' conjecture is still open when dim(i?) > 3. The rise of algebraic i 5, then an h-cobordism [W, M , M ' ) is diffeomorphic to the product M x l if and only if the Whitehead torsion T(W, M ) 6 Wh(n\M) vanishes. Moreover, Whitehead torsion gives a 1-1 correspondence between the elements of W h ( n \ M ) and diffeomorphism classes of h-cobordisms ( W , M , M ' ) of M . If ж is finite, Bass had proven in [Ba04j that W h ( i r ) is finitely generated, and computed its rank. It was implicit i n Higman's paper [H40] that the torsion i n W h ( i r ) comes from S L ( Z w ) . (Higman worked with abelian тг; the general result came much later [W74].) In spite of a heroic effort by several people in the mid1960's, examples of nonzero torsion proved to be elusive (see [MiOO, p. 416]), and not actually confirmed until the 1973 paper [ADS] by Alperin, Dennis and Stein. We refer the reader to Oliver's book [088] for the rest of this intricate story. A n important application of KQ was found by С. Т. C. Wall in 1963, and published in [W05 . If X is a space dominated by a finite complex, he defined a generalized Euler characteristic %(X) i n KQ(ZW). Wall showed that X is homotopy equivalent to a finite complex if and only if x ( X ) = 0. Naturally, we now call %(X) Wall's finiteness obstruction. L. Siebenmann [SioOOj and V . Golo [Golo] soon showed that similar obstruc­ tions exist i n K ( Z w ) to the problem of putting a boundary on an open manifold. One of the most important if-theory problems in this era was to compute the group KQ(ZW) containing these obstructions. When ж is abelian, this was 2

Q

CHARLES A. W E I B E L

8

accomplished by Bass and Murthy in [BaMur . Their idea was to decompose ж as the product of a finite group жо and a free abelian group T and then write Ъж = A [ T ] with A = Ъж$. If В is the normalization of A , this could then be analyzed by extending Rim's conductor square sequence to K\ and studying the units of B / I { T ) . When ж = тго xi Z is a semidirect product with Z , Farrell and Hsiang [FH70] found a general formula for К{Еж and W h ( n ) , generalizing the Bass-Murthy for­ mula.

The Congruence Subgroup Problem The other important application of if-theory i n the mid-1960's was the con­ gruence subgroup problem. Let R be the ring of integers in a global field F . If J is an ideal of R , let S L ( I ) denote the kernel of S L ( R ) —f S L ( R / I ) . Since R / I is a finite ring, S L ( I ) has finite index in S L ( R ) . The congruence subgroup problem asks if every subgroup Г of finite index contains a subgroup S L ( I ) for some I , so that it corresponds to a subgroup of the finite group S L ( R / I ) . One restricts to n > 3 because it is classical, and known to Klein in the last century, that there are many other subgroups of finite index in 5 X ( Z ) . If n > 3, we see if-theory appearing: the answer to the congruence subgroup problem for S L ( R ) is 'yes' if and only if S K i ( R , J) = 0 for every nonzero ideal I of R . Indeed, for any nonzero I the group E ( R , I ) also has finite index, and Bass' stability theorems in [Ba64] show that the quotient S L ( I ) / E ( R , I ) is isomorphic to the subgroup S K i ( R , I ) of K \ ( R , I ) . Conversely, if Г с S L ( R ) is normal, has finite index and n > 3, Bass used stable range conditions for R to observe in [Ba64] that there is a unique ideal I such that E ( R , I ) С Г С S L ( I ) . For R = Z , Mennicke [Me65] and Bass-Lazard-Serre [BaLS] proved that the congruence subgroup problem has an affirmative answer. Gradually the general problem was brought into focus. First Mennicke considered the coset classes of the n

n

n

n

n

n

n

2

n

n

n

n

n

n

n

subgroup бХг ( R ) of 5Хз ( R ) , modulo E3 ( R ) . The class of a 2 x 2 matrix only depends upon the first row (a b) of the matrix, and Mennicke showed that this class is multiplicative i n a and b. This led Milnor to introduce the universal Mennicke group generated by symbols [*], where b 6 I and a 6 1 + 1 are relatively prime, subject to Mennicke's two relations; this universal group maps to SK\ ( R , I ) by sending the Mennicke symbol [*] to the class of a 2 x 2 matrix with first row (а b). Milnor observed that this could be used to bound S K i ( R ) when R is a ring of algebraic integers. Then K u b o t a used power residue symbols to produce charac­ ters on congruence subgroups (in the totally imaginary setting). Bass generalized Kubota's arguments, using stability theorems for K \ , to prove that the universal Mennicke group is isomorphic to SK\ ( R , I ) for every Dedekind ring R . The congruence problem was finally settled for all global fields i n the 1967 paper [BaMS] by Bass, Milnor and Serre. The answer is 'yes' if and only if F has an embedding into ffi. If there is no such embedding, then for each ideal I they give a formula for a number m so that the group ( x of m t h roots of unity lie in F and m

THE

SK\(R,I)

DEVELOPMENT OF ALGEBRAIC if-THEORY

= j i . Under this isomorphism, the Mennicke symbol [*] m

9

corresponds

to Hasse's classical power residue symbol (|) . The solution revealed an unexpected interaction between the relative groups K \ ( R , I ) and explicit power reciprocity laws in F. This was to attract the attention of many people to if-theory, including H . Matsumoto [Mat68] and John Tate [Ta70].

Classical consolidation (1966-1968) During the next few years, the theory was reorganized, consolidating many ideas into a few structural results. This would set the stage for the discovery of K , and form the backdrop for the appearance of the higher if-theory in the next few years. The first book to appear was based on Bass' Tata lectures [BaRoy], written in 1966. Starting with a symmetrical monoidal category S, the relationship between KQ ( S ) and K i ( S ) was systematically exposed. On the next page, we shall say more about the applications to S = A z and S = Quad in [BaRoy]. Bass and Murthy's paper [BaMur] also upgraded a lot of machinery into a form suitable for calculations. For example, they clarified the Heller-Reiner inter­ pretation of the localization sequence for A —f S ^ A using the category M.s(.4) of 5-torsion A-modules. For an integral extension .4 —> В with conductor ideal J , they showed that excision holds for KQ i n the sense that KQ ( A , I ) = K Q ( B , I ) . This allowed them to extend Rim's Mayer-Vietoris sequence back as far as K\ ( A ) . They also proved that K \ ( A , I ) = K \ ( B , I ) when В is a finite product, each factor of which is a quotient of A . 2

Swan's 1968 book [Swan], based on a Fall 1967 Chicago course, gave a very readable account of the theory, focussing on KQ and K\ of abelian categories and their additive subcategories. The lecture notes of Milnor's 1967 Princeton course were distributed widely, although they were not published until 1971 as [Mi71 . They contained a clear account of the Mayer-Vietoris sequence based upon Milnor's notion of patching projective modules. Swan's book [SwEv] was also written in this period, and appeared in 1970. This book summarized the practical applications of algebraic if-theory to the represen­ tation theory of finite groups and orders, including the new induction techniques from Lam's thesis [L08 . However, the focal point of this era was the appearance of Bass' comprehensive book [Bass]. Based upon a 1966/1967 Columbia University course, and largely written in the summer of 1967, it provided a unified exposition of the subject. It also included Lam's induction techniques [L68] as well as an extended discussion of the relation between K\ of a Dedekind ring and reciprocity laws. B u t most importantly, it contained a proof of the "fundamental theorem" for K \ : the BassHeller-Swan map д and its splitting fit into a split exact sequence: 0 -»• K \ ( A )

Кг(АЩ) e K i i A l t - ] ) 1

-»• tfi(A[M ]) A K Q ( A ) -)• 0. -1

This sequence first appeared i n [BaMur, 10.2], except that exactness at the sec­ ond term was not established there. Bass and Murthy gave a similar formula for

C H A R L E S A. W E I B E L

10

K Q ( A ) = Pic(A) when A is 1-diniensional. It was further analyzed in [Swan], but the entire theorem was not made precise until [Bass]. In fact, the fundamental theorem gives a definition of the functor KQ as a function L F of the functor F = A\. Bass recognized that this process could be iterated to define negative A"-groups A"_„(i?) as Ь Ко(Я) [Bass, p. 677]. To axiomatize the process, Bass introduced the notion of a contracted functor; the fact that K i is contracted implies that K is also contracted. Independently, Karoubi [Ka68] gave a definition of -fT-groups K A ( n > 0) for any (idempotent complete) additive category A , and proved that his K P ( R ) agree with Bass' K - ( R ) . п

Q

n

N

N

Quadratic Forms and A-theory The study of quadratic forms has a long and venerable history, going back to Gauss, and focussed on the W i t t group W ( k ) of a field k, introduced by E . W i t t in [Wi37]. Classical invariants include the discriminant (in k / k ) and the Hasse invariant, lying in the Brauer group B r ( k ) . In the mid 1960's, the idea of replacing GL by the orthogonal and unitary groups led to the "hermitian" A"-theory of hermitian forms (and/or quadratic forms), and this evolved into L-theory. x

x 2

n

The algebraic part of the story begins in 1964, when Wall [W64] generalized the Hasse invariant to one lying in a larger group we now call the Brauer-Wall group B W ( k ) . This was generalized to any commutative ring R in Bass' Tata lectures [BaRoy]. Bass first formalized the notion of the category Quad(i?) of quadratic forms over R and their isometries, and considered the groups /\,Quad(/» ). He observed that the W i t t ring W ( R ) of quadratic forms is the quotient of A"oQuad(i?) by the image of a hyperbolic map K Q ( R ) —> A"oQuad(i?). Bass also showed that K\ Quad (Я i is related to the stable structure of the orthogonal groups 02n(-R). >

Next, Bass introduced the category A z ( R ) of Azumaya algebras over a commu­ tative ring R , and studied A", Az(/?)- The Brauer group B r ( R ) , defined by M . Auslander and 0 . Goldman in 1960, is a natural quotient of K Q A Z ( R ) . Bass defined the Brauer-Wall group B W ( R ) as the corresponding quotient of K Q A Z 2 ( R ) , where A z is the graded analogue of Az. Finally, he showed that the Clifford algebra functor Quad(Яi —f A z induces Wall's invariant W ( R ) —*• B W ( R ) . The topological part of the story comes from surgery theory. Suppose M is a closed manifold with fundamental group ж. The surgery problem asks when a map ф: M —> X can be made cobordant to a homotopy equivalence. For simply connected M , this problem was solved by Milnor, Kervaire, Browder and Novikov; one needs to lift the class of the normal bundle of M from K ( M ) to K ( X ) . In order to solve this problem when ж is non-trivial (and X is nice), С. Т. C. Wall [W66] observed that the middle homotopy group of ф is a projective Zir-module, equipped with a hermitian form Q. Wall then defined a 4-periodic sequence of Grothendieck groups of such forms; these groups were quickly christened L ( n ) , or Ь (Ъж). Wall then proved that the obstruction to the surgery problem for ф is the class of Q in L ( n ) , where m = d i m ( M ) . Motivated by the topological applications, the subject rapidly evolved. Stability theorems were given by B a k [B69] and Bass' foundational paper [Ba73], which discussed A'oQuad' (/?) and A', Quad' (/?) for a ring with involution R and e 6 R 2

2

A

A

m

т

m

THE

D E V E L O P M E N T OF ALGEBRAIC if-THEORY

11

such that ее = 1. It turned out, however, that Wall's 4-periodic obstruction groups L ( R ) are subquotients of these groups [W73, p. 286], with L ( R ) = L^ (R). Much of the development of this subject was due to topologists, including Wall, Novikov, Hsiang, Cappell-Shaneson, Bak, Browder, and Ranicki. In 1970, when Quillen's plus construction for higher if-theory was discovered, it was natural to consider the homotopy groups of the space B O ( R ) as the H e r m i t i a n if-theory of R ; see [Ka73]. However, it soon became apparent [Lo76] that for m > 2 these groups are less well related to the Wall groups L ( R ) . A t this point these two approaches became disconnected from each other, and also from the development of algebraic if-theory. So let us return to our story. e

e

m

m + 2

n

e

e

K

2

e

+

m

arrives (1967-1971)

Steinberg [St62] gave a presentation of a universal central extension Д of a Chevalley group G over a field k, noting that the center of Д was a "multiplier" in the sense of Schur. When G = S L , we now write S t ( k ) for Д and call it the Steinberg group. Steinberg also introduced 2-cocycles k ® k —f S t ( k ) (p. 121); we now write these cocycles as { x , y } and call them Steinberg symbols. It was gradually realized that Steinberg's presentation gave a universal central extension S t ( R ) of the subgroup E ( R ) of G L ( R ) for any associative ring R (if n > 5). n

n

x

n

n

x

n

n

As we have mentioned, the Bass-Milnor-Serre paper [BaMS] revealed an un­ expected connection between the relative groups K\ ( R , I ) and explicit power reci­ procity laws in the field of fractions for R . For example, if R is a ring of integers in a number field, they observed that there is a central extension 1 —>• K \ ( R , I ) —>• ^ ^ —>• E ( R / I ) —>• 1. E (R,l) n

n

In Spring 1967, Milnor gave a course at Princeton University, in which he defined K of a ring and developed several of its basic properties. Writing S t ( R ) for the direct limit of the groups S t ( R ) , Milnor defined K ( R ) to be the kernel of the homomorphism S t ( R ) —f E ( R ) . Milnor also wrote a letter to Steinberg explaining his results on K ( R ) . The contents of his letter were included in Steinberg's Yale course notes on Chevalley Groups [St67, pp. 92ff.], distributed in Fall 1967. 2

n

2

2

Milnor's course notes themselves were not published for several years, because of the unprecedented explosion in mathematics that they produced. When the book [Mi71] finally appeared, it incorporated many of the results found during 1967-1970. First came the work of C. Moore [Mor68] (written in 1967). If F is a local field then the norm residue symbol maps K ( F ) onto the finite group ц = ц of roots of unity in F ; Moore showed that its kernel V is divisible, so K ( F ) = цр ®V. In fact, V is uniquely divisible; this was proven much later by Tate [Ta77] when char(F) = p, and by Merkurjev [Me83] when char(F) = 0. Next, Matsumoto's 1968 thesis [Mat68] gave a presentation of K ( F ) when F is a field: it is generated by the Steinberg symbols { x , y } , which are subject only to bilinearity and the relation { x , 1 — x } = 1 (which was discovered by Steinberg). 2

п

2

2

CHARLES A. W E I B E L

12

Let R be a Dedekind domain with field of fractions F. For each prime ideal p of R , Bass had constructed transfer maps K \ ( R / p ) —f S K \ ( R ) i n [Bass, p. 451]. In 1969, Bass and Tate constructed the tame symbols A : K ( F ) —> K i ( R / p ) and used reciprocity laws for A = фА similar to those i n [BaMS] to construct the exact sequence p

2

р

K (R)

K (F)

2

2

A (BpK^R/p)

S K ^ R )0

[Mi71, p. 123] [BaT . Their proof of exactness at K ( F ) required R to have only countably many prime ideals, but that suffices for integers i n global fields. When R is the ring of integers i n a number field F, the kernel of A was called the tame kernel until 1972, when Quillen discovered [Q73] that the tame kernel was just K ( R ) . In 1971, Garland [Gar71] proved that the tame kernel K ( R ) is finite. Since Tate was able to calculate the tame kernel for F = Q as well as the first six imaginary quadratic number fields [BaT], K ( F ) was known for these fields. 2

2

2

2

In 1970/71, J . Tate [Ta70] discovered the norm residue symbols on K ( F ) (one for each n with 1/n S F ) . If F has primitive n-th roots of unity, they give maps K ( F ) —f B r ( F ) ; i n general they take values in the etale cohomology group B~ (F,fx® ). Tate proved that K ( F ) / n ->• H (F,ii® ) is an isomorphism for all global fields F, but the details were not published until several years later, i n [Ta76]. The proof that this map is an isomorphism for all fields was discovered in 1981/82, by Merkurjev and Suslin [Me81] [MS82]. 2

2

2

t

2

2 t

2

2

Birch [Bir69] and Tate [Ta70] conjectured that if R is the ring of integers i n a totally real number field F then the order of the finite group K ( R ) should be a certain number w ( F ) times the value (F(—1) of the Riemann Zeta function of F. The odd part of this conjecture was settled by A . Wiles i n [Wi90]; the 2-primary part of this conjecture is still open, and is equivalent to part of the (2-adic) M a i n Conjecture of Iwasawa Theory; see [RW97, А.1]. 2

2

In the 1970 paper [Mi70], Milnor observed that the Hasse invariant of qua­ dratic forms over a field F could be factored through Tate's norm residue symbol K ( F ) / 2 —f B r ( F ) . More precisely, if I denotes the kernel of W ( F ) —> Z / 2 then the Hasse invariant maps I to B r ( F ) , and Milnor proved that it factors through an isomorphism J / / = K ( F ) / 2 . This led him to introduce what we now call the M i l n o r К-groups of a field F, v i z . , a graded ring K ^ f ( F ) which agrees with K * ( F ) for * = 0,1,2. Milnor constructed a canonical map from K f f ( F ) / 2 onto I / I for each n , proved that it is an isomorphism for local and global fields, and asked (p. 332) if it is an isomorphism for every field F. If char(F) = 2, this was solved positively by K . Kato in 1981 [К82]. 2

2

2

3

2

n

n

+

1

The Milnor ring K ^ f ( F ) of a field was systematically studied by Bass and Tate in [BaT]. For global fields, they were able to compute K ^ ( F ) = ( Z / 2 ) for all n > 3, where r\ is the number of embeddings (if any) of F into Ж. They also constructed transfer maps K ^ ( E ) —f K ^ ( F ) for certain finite field extensions F с E; the proof that transfer maps exist for all finite extensions was given i n 1980 by Kato [К80]. After K a t o distributed his 4-page proof at the 1980 Oberwolfach conference, Suslin immediately used Kato's transfer maps to construct a map K ( F ) —f K ^ f ( F ) , and prove that the natural map K ^ f ( F ) —f K ( F ) is an injection modulo torsion [Su80]. ri

N

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THE

D E V E L O P M E N T OF ALGEBRAIC if-THEORY

13

Milnor also constructed higher norm residue symbols K™(F)/2 H ? ( F , Z/2) in [Mi70], and used Tate's results to prove that h is always an isomorphism for local and global fields. Milnor then stated (p. 340) that he did not know of any examples where it fails to be an isomorphism; this became known as the "Milnor conjecture" after the case n = 2 was settled in 1981 by Merkurjev [Me81]. Jump­ ing ahead, the case n = 3 was solved in 1986 by Merkurjev-Suslin [MS90] and M . Rost. A proof that h is an isomorphism for all n was only discovered recently by V . Voevodsky [Voe96]. t

n

n

For rings other than fields, progress was slower. First, the group K ( R [ e ] ) f K ( R ) was computed by Wilberd van der Kallen [vdK71 . Then Dennis and Stein [DS73] [DS75] constructed symbols (a, b) in K ( R ) and showed that they form a complete set of generators for K of a semilocal ring. If J is a radical ideal, they also showed that K ( R , I ) is generated by the Dennis-Stein symbols (a, b) with о or b in I . A full set of relations for the Dennis-Stein symbols, and hence a presentation for K ( R , I ) was later given by Maazen and Stienstra in [MaSt]. 2

2

2

2

2

2

Higher Ж-theory arrives (1968-1972) The search for higher if-groups had dominated much of the 1960's, as we have seen. It was clear that Milnor's definition of K should form the basis of this search. In fact, Swan proposed the definition K ( R ) = H ( S t ( R ) ; Z ) in 1967 [Swan, p.207]. In 1968, Swan [Sw68] proposed a definition of higher groups K ( R ) , based upon the idea that free rings should have no higher if-theory. In 1969, Steve Gersten [Ger69] used the free ring functor F to produce a functorial "cotriple" resolution F * R of any ring R ; applying the functor R »->• E ( R ) gives a simplicial group, and Gersten defined K ( R ) = n - E ( F * R ) for n > 2. Swan [Sw70, Sw72] showed in 1970 that his definition agrees with Gersten's. In 1968, Nobile and Villamayor [NV68] gave a definition of higher if-theory, essentially by defining the "loop space" of a ring. These ideas led Karoubi and V i l ­ lamayor to define groups K ^ ( R ) in [KV69] (whose details appeared in [KV71]). The drawback is that their group K ^ ( R ) is the quotient of K\ ( R ) = G L ( R ) / E ( R ) by the subgroup generated by the unipotent matrices, but it has the advantages that K ^ ( R ) = K ^ ( R [ t ] ) for every n > 1, and that long exact sequences (such as Mayer-Vietoris sequences) exist in many situations. Nowadays, we write K V ( R ) for K ^ ( R ) and call it the K a r o u b i - V i l l a m a y o r if-theory of R . Gersten [Ger70] gave a simpler version of Karoubi-Villamayor theory in 1969, using the notion of a homology theory associated to the functor G L from rings to groups. Then Rector remarked [Glet] [R71, 2.6] that to every ring R one could associate a simplicial ring A R with A R = R [ t o , . . . , t ] / ( J 2 i i = 1)) and showed that the homotopy groups of the simplicial group G L ( A R ) also give the KaroubiVillamayor groups. Rector's remark has formed the basis of all subsequent work in the subject. Historically, the most important construction of higher if-theory was the plus construction, given by Quillen in 1969/70. In his work [Q$] [Q70] on the Adams Conjecture, Quillen constructed a map from BGL(¥ ) to the fiber F $ of the 2

3

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1

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n

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?

14

CHARLES A. W E I B E L ф —1 ч

Adams operation B U > B U for each finite field F , and observed that the map is acyclic (induces isomorphisms on homology). Quillen observed more generally that if one started with any C W complex X then there is an acyclic map X —f X , unique up to homotopy, such that щ ( X ) —f щ ( X ) is the quotient by the largest perfect subgroup. In particular, BGL(¥ ) has the homotopy type of F $ . Motivated by this, Quillen defined K ( R ) as the homotopy group w B G L ( R ) [Q71, p. 50]. Since the homotopy groups of F $ could be read off from the known action of ip on B U , this gave the calculation of K (¥ ) for every finite field F . Moreover, i n light of the Kahn-Priddy theorem that B H ~ O ^ , the inclusion of the symmetric groups E„ i n G L ( Z ) induces maps тг* —f K ( Z ) . Quillen showed in [Qlet] that these maps induced an injection of the cyclic groups J ( w O ) С тг* into K ( Z ) when n = 3 (mod 4). Also in 1970, G . Segal was developing his infinite loop space machine which starts with a symmetric monoidal category S (or more generally a Г-space) and produced an fi-spectrum. (His paper [Seg74] only appeared i n 1974.) Quillen showed in the 1971 preprint (now published as [Q94 ) that if one takes S to be the category of fin. gen. projective i?-modules and their isomorphisms then Segal's infinite loop space is K Q ( R ) X B G L ( R ) . Of course, the same could be done with other infinite loop space machines, such as May's [May72 . This choice caused some confusion until 1977, when May and Thomason showed in [MaThj that all reasonable infinite loop space machines agree. Several other constructions of K ( R ) appeared i n 1971. One which was based upon the theory of buildings and upper triangular subgroups was given by I. Volodin [Vo71 . In an attempt to understand Volodin's construction, J . Wagoner [Wag73] came up with a similar construction. A construction along the lines of Swan's was given by F. Keune [Kou71 . In 1972, Quillen [Q72, Q73] defined the higher if-theory of an exact cat­ egory, viz., a full subcategory A of an abelian category which is closed under extensions. To do so, he first constructed an auxiliary category Q A and de­ fined K ( A ) = w ^ i B Q A . Theorem 1 of the announcement [Q72] stated that the loop space Q B Q P ( R ) is homotopy equivalent to K ( R ) x B G L ( R ) , so that K ( R ) = K P ( R ) . Awkwardly, the proof of this "+ = Q " theorem was not pub­ lished until 1976 [GQ76]. g

+

+

?

+

q

n

+

n

?

q

n

g

q

0 0

+

n

0 0

n

n

n

+

n

n

n

+

Q

n

n

Theorems 2-4 of [Q72] extended the fundamental structure theorems from classical if-theory ( K for n < 2) to higher if-theory: devissage, resolution and localization. Theorem 5 stated that if R is regular then K * ( R ) = G*(R), where G*(R) is the if-theory of the abelian category of finitely generated R-modules. Theorem 8 was the localization sequence for a Dedekind domain, and theorem 11 was the Fundamental Theorem: N

G ( R [ t ] ) s G (R) n

n

and С ( Д [ М ] ) - G„(R) Ф П

- 1

G„-i(R)-

Now Gersten had given a spectral sequence [Glet] from K * ( A R ) to the KaroubiVillamayor groups K V * ( R ) . If R is regular, Quillen's Fundamental Theorem im­ plies [Q72, p. 101] that Gersten's spectral sequence degenerates, giving K„(R) = K V * ( R ) . Quillen concluded by defining the groups G * ( X ) of a quasi-projective

THE

DEVELOPMENT OF ALGEBRAIC if-THEORY

15

scheme X , and giving a spectral sequence converging to G* ( X ) , starting from the .FT-groups of the residue fields of X . A t this point in the rapid-fire sequence of events, Bass organized a two-week conference on algebraic iT-theory. Held at the Battelle Memorial Institute in Seattle during August 28-September 8 of 1972, the relationship between the various K theories became one of the focal points of the conference. As we have seen, things had been greatly clarified by Quillen's announcement [Q72 . Gersten used Roberts' thesis [Rob76] to show [Ger72] that the Bass-Heller-Swan group K ( A ) differs from Quillen's K \ ( A ) when A is the category of vector bundles over an elliptic curve. The consensus at the Battelle Conference was that Quillen's iT-theory of a free ring Z { X } measures the difference between the different constructions. A few months afterward, Gersten proved [Ger74] that i T , ( Z { X } ) is trivial. Using this, Don Anderson proved in [A73] that Quillen's groups agree with the Gersten-Swan groups, and Wagoner showed that they also agree with his. A proof that Volodin's construction agrees with these was not given until much later, by Suslin [Su81]. Encouraged by Bass, Quillen wrote the foundational paper [Q73] during the first two months of 1973. It was a masterful piece of exposition, providing tools for calculations and elegant proofs of the structure results announced i n [Q72 . (We will not elaborate on the details in order to focus on the flow of events.) A t that time, Quillen discovered proofs of Gersten's conjecture and Bloch's Formula; we will discuss these results below. X

Applications to Algebraic Geometry The interaction of higher iT-theory and algebraic geometry begins with K e n Brown's 1971 M I T thesis [Br73], which developed the cohomology theory of sheaves of spectra on a noetherian scheme X . Brown was motivated by Quillen's sug­ gestion that such a theory might apply to iT-theory. Gersten had recently con­ structed a functorial spectrum K ( R ) associated to a ring R ; sheafifying gives such a sheaf spectra. Seizing upon this, Gersten announced i n [Ger73] that if we set K ( X ) = H ( X , K) for regular separated schemes X then for X = Spec(i?) we re­ cover K ( X ) = K V ( R ) . (Quillen had already announced that K V ( R ) = K ( R ) [Q72], but Gersten was unaware of this at the time.) Moreover, there is a spec­ tral sequence converging to K * ( X ) whose -terms are the Zariski cohomology H ( X , K . g ) of the sheaves associated to the presheaves sending Spec(i?) to K ( R ) . The details were worked out by Brown and Gersten at the Battelle Conference, and appeared in [BG73 . n

n

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n

p

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Jouanolou [J73] also extended the functors K V from rings to quasi-projective schemes using what we now call "Jouanolou's device:" for each such X there is a ring R and a map Spec(i?) —f X which is locally a vector bundle; one defines K V * ( X ) = K V * ( R ) . A t the Battelle Conference it became clear that Quillen's def­ inition K ( X ) = A'„P(.V i agreed, at least for regular X , with the groups obtained using Jouanalou's device, as well as the Brown-Gersten definition [BG73, Q73 . n

n

In the other direction, the application of higher iT-theory to the study of cycles in algebraic geometry was initiated by Spencer Bloch in [В174]. Bloch's preprint

CHARLES A. W E I B E L

16

appeared just before the 1972 Battelle Conference, and was influenced by Gersten's results i n [Ger73]. His main theorem, that the cohomology group H ( X , K . ) is isomorphic to the Chow group C H ( X ) on any regular surface, is now called Bloch's formula. One interesting conjecture made in [B174] was that if R is a (regular) local domain with quotient field F , then the following sequence is exact: 2

2

2

0 -»• K ( R )

K ( F ) A ® =ik(p)

2

2

x

Up

-»• e -»• 0, 2

where k ( p ) denotes the residue field of R , and C is closely related (see p. 360) to the free abelian group on the height two prime ideals of R . Gersten seized upon Bloch's formula, and made a series of striking conjectures about the higher iT-theory of regular local rings, including what we now call Ger­ sten's conjecture: K ( R ) injects into K ( F ) for every regular local ring R . These conjectures were presented at the Battelle Conference in 1972. When writing up his foundational paper [Q73] in early 1973, Quillen discovered a proof of Gersten's Conjecture for the local rings of a nonsingular variety X over a field. His proof implied that Bloch's formula generalized to H ( X , K ) — C H ( X ) for all p . Quillen included this material in [Q73]. Looking ahead, Gersten's conjecture was proven for equicharacteristic D V R ' s by C. Sherman in [Sh78]. The general mixed characteristic case of Gersten's conjecture is still open today, i n spite of some progress in the 1980's by Bloch, Gabber, Gillet, Levine, and Sherman. Bloch-Ogus [BlOgs] soon showed that Quillen's trick for solving Gersten's conjecture could be used i n other settings, such as de Rham cohomology. Gillet [Gi80] later used the ideas i n the Bloch-Ogus paper to define Chern classes for higher iT-theory, and to prove a very general Riemann-Roch theorem. p

n

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P

P

In 1978, Bloch [B179] used the cohomology of the sheaves K to describe the Abel-Jacobi map C H ( X ) —f A l b ( X ) of a smooth projective n-dimensional variety, and to give a different proof of Roitman's theorem that the Abel-Jacobi map induces an isomorphism between their torsion subgroups. In the 1980's, the use of K cohomology to study cycles exploded, starting with [CSS82], but that is beyond the scope of this paper. The state-of-the-art as of 1979 is nicely described in Bloch's book [В180]. n

n

Homological methods Most attempts to compute the groups K ( R ) in the 1970's used homological methods. The idea was to use geometric methods to calculate either the homology of the groups S L ( R ) or the homology of B Q for small subcategories Q of Q P ( R ) . If R is the ring of integers i n a number field, Quillen [Q73Z] used these methods to show that the K ( R ) are finitely generated, and Borel [Bo72] used them to compute the rank of the K ( R ) . Motivated by these calculations, and also by his calculations i n [Ba64], Bass posed this question at the Battelle conference: are the groups K ( R ) finitely gen­ erated whenever R is either finitely generated as an abelian group, or regular and finitely generated as a ring? This question has become known as Bass' "finite generation conjecture," and is still open. n

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THE

DEVELOPMENT OF ALGEBRAIC if-THEORY

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Here are two of the strongest results in this direction. If R is regular, finitely generated as a ring, and dim (12) < 1, Quillen proved that the K ( R ) are finitely generated in 1974; his proof was published in [GQ82 . If R is finite, or R is an order in a semi-simple Cj-algebra, Remi K u k u [Ku86] proved that each K ( R ) is finite. Several people used homological methods to prove stability theorems for higher .FT-groups, including Charney, Maazen, Vogtman, van der Kallen and Suslin. n

n

It was sometimes possible to use homological methods to calculate i T and even i T . In 1975, Lee and Szczarba [LeeSz] were able to calculate the homology of 5 X ( Z ) and prove that -FT (Z) is isomorphic to the cyclic group Z/48. Evens and Friedlander computed i T and K\ for rings of order p in [EF80]. However, the dif­ ficulty of using this method quickly became overwhelming, as Staffeldt's 1977 thesis [St79] shows. After a ten year effort by V i c Snaith and his students culminated in the book [ALS85], the effort to use these techniques was effectively abandoned. 3

4

3

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2

3

The Lichtenbaum Conjectures Let F be a totally real number field F, and R its ring of integers. In 1972, S. Lichtenbaum [1Л73] conjectured that the iT-theory of R is related to the values of the zeta function ( F ( S ) - More precisely, for each % > 1 he conjectured that the rational number \ ( p ( l — 2i)\ equals the ratio of the orders of the finite groups i T _ ( i ? ) and K ^ ( R ) . The first test of this conjecture was R = Z and i = 1. Since CQ(—1) = —1/12 and i T ( Z ) = Z/2, Lichtenbaum's conjecture predicted that i T ( Z ) has 24 ele­ ments. After Lee and Szczarba [LeeSz] showed that it has 48 elements, Lichten­ baum amended his conjecture to say that equality holds "up to a power of 2." W i t h hindsight, we know that the connection between the iT-theory of R and the zeta function of F goes through the etale cohomology of R. To explain this connection, note that the Galois group G of F / F acts on the roots of unity ду and its twists For each % there is an integer w,(F) such that the groups R~% (F, (xf/) are cyclic of order w,(F) for all large N. We have already seen that the number u>2 appears in the Birch-Tate conjecture that #iT (i?) is w (F)|£ (—1)|. Another connection was found by Bruno Harris and Graeme Segal, who proved in [HS75] that i T , _ i ( i ? ) contains a cyclic summand whose order is w ( F ) up to a factor of 2. Lichtenbaum had already made several conjectures in number theory, including a conjecture that for all % and all primes £, the €-part of the integer W i ( F ) ( p ( i — 1) equals the order of Hl (R[l/£], J ) t o r s for all large v. In 1990, Wiles [Wi90] proved the M a i n Conjecture of Iwasawa Theory for odd primes £, which implies this conjecture up to powers of two. The 2-part of this conjecture would follow from the M a i n Conjecture for the prime 2, which is still open. Comparing these two circles of ideas, we may rephrase the second half of Lichtenbaum's conjectures as saying that for v large, we have 4j

2

U

2

3

2

2

2

2

t

„(?)

#K '

2

I

#H

-2(R) (Д)

П

2

t

(R[im^fj)

#Hl (R[l/l];vfJ) t

F

C H A R L E S A. W E I B E L

18

We now know [RW97] that "(?)" is the number п of real embeddings of F. In fact, thanks to the work of Voevodsky in [Voe96], Rognes and Weibel [RW97] have recently affirmed the 2-primary part of the second half of Lichtenbaum's conjectures. Quillen [Q75] generalized this second half, suggesting that for any regular ring R there should be a spectral sequence for each prime £ invertible in R , analogous to the Atiyah-Hirzebruch spectral sequence, but starting from the etale cohomology groups H f ( R , Z|(g/2)), whose abutment would coincide with the £-adic completion of K ( R ) at least in degrees n > 1 + dim(i?). If R is the ring of integers in a number field (and £ is odd), such a spectral sequence would degenerate, yielding the cohomological formulas for K ( R ) conjectured by Lichtenbaum. Quillen made this conjecture explicit when F is an algebraically closed field of characteristic p: the groups K 2 ( F ) should be uniquely divisible, and i T _ i ( F ) should be a divisible group whose torsion subgroup is Q/Z[l/p]. This was affirmed by Suslin in 1983 paper [Su83]; the case p = 0 was given in the sequel [Su84]. During the late 1970's, it gradually became clear that Quillen's conjectures should be attacked using iT-theory with finite coefficients. The groups K*(R;Z/£) were first introduced by W . Browder [Br76], who constructed products on them and showed that the Bott element /3 6 iT (i?; Z/£) plays a central role. For example, if a finite field F has q = 1 (mod £) then K*(F ;Z/£) is the graded commutative ring Z/£[x,f3]/(x = 0), where x 6 K\ is the class of a unit generating F * . t

n

n

2 n

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2

g

q

2

Soule [Sou79] constructed etale chern classes K ( R ; Z / t ) ->• H p ( R , f i f j ) , and showed that they are onto in many cases. To do this, he established a product formula, and applied it to elements in iT* ( R ; Z /£) of the form f3 Ux with x 6 K j ( R ) . Soule showed that the orders of these elements are related to the numerator of the Bernoulli numbers B ^ / i k , as predicted by Lichtenbaum's conjectures. For example, Soule produced an element of order 691 in i T ( Z ) . Inspired by Soule's calculations, Eric Friedlander invented an approximation to iT-theory he dubbed "etale iT-theory." Dwyer and Friedlander [DF82] constructed a spectral sequence of the kind described by Quillen, converging to etale iT-theory. Later on, it was proven that etale iT-theory of X is obtained from the ring K ( X ; Zj£ ) by inverting the Bott element, and that K [X;Zj£ ) agrees with etale iT-theory for large n . The proof of these assertions was published in [DFST] and also in Thomason's masterpiece [Th85]. Backing up somewhat, we turn to the question of regulators. Let R be the ring of integers in a number field F. Borel had observed in [Bo72] that the rank dj of the group i T j - i ( i ? ) equals the order of the zero of the zeta function ( F ( S ) at s = 1 — % for each i. Indeed, the Hurewicz map iT,(i?) —f H * ( G L ( R ) ; Ж) induces a map r , : i T , _ i ( i ? ) —f R , which we now call the Borel regulator. In 1976, Borel showed [Bo77] that the image of r, is a lattice whose covolume is a rational number times 2

n

n

l

22

v

n

v

n

2

d i

2

n-

d i

lim ( ( s ) ( s + i - l ) F

d

i

=

TT-^ ^VDC (i). F :

F

s—>1 — i

Here di is the order of the pole of the zeta function ( F ( S ) at s = 1 — i, and is either r or r\ + r depending on % (mod 2). Next, Bloch gave an intense series of lectures on Borel's regulator at the Uni­ versity of California at Irvine, followed by some conjectures about possible general­ izations to regulators for an elliptic curve E over a number field. Writing up these 2

2

THE

D E V E L O P M E N T OF ALGEBRAIC if-THEORY

19

ideas led him to discover a regulator K z i E ) —f R , which was included in the 1977 preprint [B177 : Bloch gave a talk on this at the 1978 Helsinki I C M . In September 1979, Manin asked an undergraduate student in Moscow to give a report on Bloch's Irvine notes [В177]. That student was A . Beilinson, who was then 22 years old. Instead of giving Bloch's construction, he improved on it in [Bei80] by constructing what we now call Beilinson's regulator. This led Beilinson to make a series of conjectures in 1982 [Bei82, Bei84]; these are now called the Beilinson conjectures, and they have been a driving force in the development of algebraic iT-theory in the 1980's and 1990's. While the Beilinson conjectures are extremely interesting, they take us well beyond 1980 in our historical account. So we restrain ourselves from continuing past this point, and turn back to other developments which took place in the 1970's. d

Symbol Calculations From the beginning, there was an interest in constructing and detecting el­ ements of higher iT-theory. It was implicit from the start [Ger72, p. 17] that Quillen's theory should have products iT,(A) ®Kj(B) —f Ki j(A®B). Such prod­ ucts were independently constructed by Loday [Lo75], using the space B G L ( R ) , and by Waldhausen [Wa73], who used an iterated Q-construction in the context of pairings of exact categories. In the published version of [Wa73], Waldhausen gave a proof that his products agree with Loday's; see [We80] for more details. Products allowed one to form higher Steinberg symbols { x i , . . . , x } in K ( R ) , yielding the ring homomorphism K ^ ( F ) —f iT„(F) discussed above. Products were used to construct other elements in K ( R ) , such as the generalized DennisStein symbols (r\, . . . , r ) , which are defined whenever 1 — FJ r, is a unit in R [Lo81 . Applying cohomology to these products gives a bigraded ring H * ( X , K * ) . Grayson [Gr77] used Waldhausen's product formalism to show that Bloch's for­ mula H ( X , K . ) — C H ( X ) is actually a ring isomorphism. Products were also used very effectively by Gillet [Gi80] in proving a version of the Riemann-Roch theorem for higher iT-theory. +

+

n

n

n

n

P

P

P

To show that these new elements were nonzero, people turned to Chern classes. Gersten had constructed Chern classes with values in Kahler differentials fl* as early as 1970 [Glet] [Ger72, p. 238]. These methods were used by Bloch in [B178] to construct Chern classes in crystalline cohomology, as part of an effort to relate iT-theory to the slope spectral sequence in crystalline cohomology. Product formulas for the etale chern classes [Sou79] and for the classes in iT* ( X , K.* ) [Su80] showed that Chern classes on K typically could not detect elements of exponent ( n - 1)!. In 1976, Keith Dennis [D76] discovered trace maps from K ( R ) to the Hochschild homology group H H ( R ) , but never published the details. These Dennis trace maps were to play a major role in calculations during the 1980's. Another aspect of the attempt to find generators and relations for K ( R ) was the search for a Mayer-Vietoris sequence for higher iT-theory associated to an "exci­ sion situation" where one had a ring map R —> S sending an ideal I of R isomorphically onto an ideal of S. Around 1970, Swan [Sw71] had made the discovery that R

n

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C H A R L E S A. W E I B E L

the relative groups K\ ( R , I ) really depend upon R , so that there was no general ex­ tension of the Mayer-Vietoris sequence to K . Swan also gave a description for the kernel of the surjection K \ ( R , I ) —f K \ ( S , I ) . After his description was improved by Vorst [Vo79], S. Geller and C. Weibel [GW83] gave a complete description of the doubly relative term i T i ( R , S, I ) , i . e . , the preceding term in the appropriate long exact sequence. The description of the relative group K ( R , I ) followed a parallel path. In his 1970 thesis [St71], M . Stein gave one candidate for K ( R , I ) , the definition used in Milnor's book [Mi71 . Swan pointed out in [Sw71] that one should re­ ally define K ( R , I ) to be the third group fitting into the natural long exact se­ quence for iT*(i?) — f K * ( R f I ) , and this relative group was a proper quotient of Stein's candidate. Keune and Loday independently found a description of K ( R , I ) in 1977 [Keu78, Lo78 . Then in 1980, a description of the doubly relative group K ( R , R / J , I ) was given by Keune [Keu80] and, again independently, by GuinWalery and Loday [GWL . 2

2

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2

Computing the iT-theory of nilpotent ideals, a problem posed by Swan at the Battelle Conference, was also an active topic in the 1970's. The calculation of K ( R , I ) was mentioned above [MaSt]. Homological methods had limited success in characteristic p, highlighted by [EF80]. If R contains Q, then each K ( R , I ) is uniquely divisible, and rational homology methods had more success, including Soule's calculation [Sou80] when R = Q[e] and climaxing in Goodwillie's theorem [Gw86] that K ( R , I ) is isomorphic to the cyclic homology group H C - i ( R , I ) . If R has residual characteristic p, Randy McCarthy [Mc97] has recently shown that K ( R , I ) is isomorphic to the topological cyclic homology group T H C - \ ( R , I ) . 2

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n

n

n

n

Pseudo-isotopies and

A(X)

Having settled the /i-cobordism question, topologists turned to the question of uniqueness of product structures o n M x J , and studied the topological group V ( M ) of pseudo-isotopies on M , i . e . , diffeomorphisms of M x I which are the identity on M x {0}; a pseudo-isotopy which preserves each level M x { t } is an isotopy. Cerf proved that if d i m ( M ) > 5 and M is simply connected then every pseudoisotopy is homotopic to an isotopy. Hatcher and Wagoner [HW73] computed WQV(M) when ж = ж\(М) is nontrivial; it is a sum of two terms, one of which is a quotient of К (Ъж) they named УУЬ, (ж). Hatcher also discovered a connection between ж\_Т(М) and K . This led him to construct a stable space W h ( M ) as the direct limit of the spaces f l V ( M x I ) and prove that if d i m ( M ) >• n one has ж Т(М) = 7 r „ W / i ( M ) . (Unfortunately Hatcher stated the stable range incorrectly, but that is not part of this story.) In 1975, Doug Anderson and W . - C . Hsiang [AH75] studied pseudo-isotopies of polyhedra, and showed that negative iT-groups appear. For example, they showed that т г Р ( 5 М ) = К^ЪжгМ) if m > 4. Loday [Lo75] introduced the assembly map H ( B G ; S Z ) —f К (Ъж), where the domain is the generalized homology of B G with coefficients in the spectrum K S for iT* (Z), and observed that when n = 2 the cokernel is the Hatcher-Wagoner group \¥11 (ж). This led Loday and Waldhausen [Wa78, p. 228] to define the higher Whitehead groups УУЬ, (ж), not in terms of W h ( X ) but as the third term 2

2

F

2

2

k

PL

п

г о + 2

0

n

п

2

п

F

L

L

THE

DEVELOPMENT OF ALGEBRAIC if-THEORY

21

in the natural long exact sequence >H (BG;SZ) n

- > K ( Z w ) - > W h ( w ) -»• H ^ ( B G ; S Z ) n

n

•••

n

The study of pseudo-isotopies was revolutionized i n 1976, when F. Waldhausen [Wa78] constructed an infinite loop space A ( X ) , functorial in X , and a natural map A ( X ) —f W h ( X ) which was later proven [Wa87] to give a decomposition: F L

A{X) ~ n

o o

S

0 0

{X ) +

x

Wh

(X).

P L

In addition, Waldhausen proved i n [Wa78] that ж*А(Вж) ® Q = K»(Zn) Q. To do this, Waldhausen first defined the algebraic iT-theory of a "ring up to homotopy," R , using a variation B G L ( R ) of Quillen's +-construction. For example, if f l ' X is Kan's simplicial loop group for X , one can form the ring up to homotopy Q [ Q ' X ] = f l S ( f l ' X ) . Waldhausen defined A ( X ) as the iT-theory space of Q [ t t ' X ] [Wa78, p. 42]. +

0 0

0 0

+

Waldhausen also gave a iT-theoretic construction of A ( X ) in [Wa78, p. 58], a construction now called Waldhausen К-theory which we will describe shortly. Here is one nice application that came out of Waldhausen's construction, show­ ing one way that P L manifold theory is simpler than smooth manifold theory. Consider the two topological groups Нотеорь(-О , d D ) and D i f f ( D , d D ) of P L homeomorphisms, resp. diffeomorphisms, of the disk which fix the boundary. It is a well-known consequence of the "Alexander Trick" that Нотеорь(-О , d D ) is contractible. In contrast, Farrell and Hsiang [FH76] used Waldhausen's results and Borel's computation of iT„(Z) ® Q, together with an analysis of the assembly map, to compute the rational homotopy groups of Diff(D", d D ) ; they are nonzero. п

n

n

n

п

n

n

In 1978-1980, Dwyer, Hsiang and Staffeldt [DHS80, HS82] used rational homotopy theory to compute ж А(Х) ® Q when щ ( X ) = 0. They stated their answer in terms of classical invariant theory. Later on, it would be recognized as a harbinger of cyclic homology: w A(X)®Q = K ( Z ) ® H C - i ( R . ) , where R . is the differential graded algebra of singular chains on O X . The integral version of this result is given in [CCGH]. Also in [Wa78], Waldhausen described a stabilization process, using the iTtheory of a simplicial ring, to construct groups K ^ ( R , M ) , called the stable iTtheory of R with coefficients i n a bimodule M . Waldhausen constructed a natural map from i T f ( R , M ) to the Hochschild homology group H H ^ i ( R , M ) and showed that it is an isomorphism for n = 1,2; Kassel [Kas80] showed that they differ for n = 3. Later, Dennis and Igusa showed in [DI82] that Hatcher's obstruction for п

n

n

n

n

T T I P ( X ) is i T f ( Z T T , M ) , where ж = ж {Х) г

and M

= ТГ (Х)[ТГ]. 2

Looking ahead, Goodwillie conjectured that there existed a construction for rings like that used to construct Hochschild homology, and that it would be homo­ topy equivalent to stable iT-theory for any ring. Bokstedt gave such a construction in [B685], calling it the topological Hochschild Homology T H H ( R ) of a ring. This played a key role in later computations such as [ В Н М ] . Finally, Dundas and M c ­ Carthy showed in [DM94] that T H H ( R ; M ) ~ K ( R , M ) for any simplicial ring R and bimodule M . S

22

C H A R L E S A. W E I B E L

Waldhausen iT-theory Waldhausen also gave a second construction of A ( X ) in 1976. A preliminary version appeared in [Wa73, p. 181ff], and the announcement was published in [Wa78, p. 58]. However, Waldhausen's foundational paper [Wa85] containing the full details only appeared much later. For this construction, he introduced what we now call a Waldhausen category, i . e . , a category A with appropriate notions of cofibrations and weak equivalences w. Given A , Waldhausen defined a simplicial category w S . A and studied the space K { A , w ) = fl\wS.A\. (The S in the notation is for G . Segal, who had earlier considered a similar construction; see [Wa73, p. 181].) Waldhausen proved that A ( X ) ~ K ( 7 l f ( X ) , h ) , where TZf(X) is the category of finite split retractions Y —> X and h denotes homotopy equivalence over X. See [Wa78, 5.7]. This of course was the original point of the construction, and many of Waldhausen's structure theorems were based on the existence of a construction like the formation of the standard mapping cylinder, satisfying a "cylinder axiom." Any exact category A is a Waldhausen category, where cofibration sequences are exact sequences and w is the class of isomorphisms. Waldhausen observed that B Q A ~ \wS.A\, so that K * ( A , w ) = n * K { A , w ) agrees with Quillen's i T , ( A ) . A third important application of Waldhausen's construction is to the category Сь(А) of bounded chain complexes in an exact category A , where w is quasiisomorphism; the classical mapping cylinder of chain complexes satisfies Wald­ hausen's cylinder axiom. In fact, Waldhausen's group Ко(Сь(А),и>) recovers the definition of K Q ( A ) given by Grothendieck in [SGA6]. Waldhausen realized early on that if A is the category of projective i?-modules then K ( C b ( A ) , w ) ~ К ( A ) ; Gillet proved more generally [Gi80, p. 256] that this formula holds for any (idempotent complete) exact category A . (Cf. [TT90, p. 279].) Gillet applied this to the category A of vector bundles on a scheme X, showing that the Chern classes in [Gi80] are local Chern classes in the sense of [SGA6].

References J . F . A d a m s , V e c t o r fields o n s p h e r e s , A n n a l s of M a t h . 75 (1962), 603-632. J . F . A d a m s , O n t h e g r o u p s J ( X ) I , Topology 2 (1963), 181-195; II, Topology 3 (1965), 137-171; I I I , Topology 3 (1965), 193-222; I V , Topology 5 (1966), 21-71. [A73] D . A n d e r s o n , R e l a t i o n s h i p a m o n g K - t h e o r i e s , Algebraic -ff-theory I, Lecture Notes i n M a t h . , v o l . 341, Springer Verlag, 1973, pp. 57-72. [ADS] R. A l p e r i n , R. K . Dennis and M . Stein, T h e n o n - t r i v i a l i t y of S K i C E [ n } ) , Lecture Notes in M a t h . , v o l . 353, Springer V e r l a g , 1973, p p . 1-7. [AH59a] M . A t i y a h and F . H i r z e b r u c h , R i e m a n n - R o c h t h e o r e m s f o r d i f f e r e n t i a b l e m a n i f o l d s , B u l l . A M S 65 (1959), 276-281. [AH59b] M . A t i y a h and F . H i r z e b r u c h , Q u e l q u e s t h e o r e m e s d o поп p l o n g e m e n t p o u r l e s v a r i e t e s d i f f e r e n t i a b l e s , B u l l . Soc. M a t h . France 89 (1959), 383-396. [AH61] M . A t i y a h and F . H i r z e b r u c h , V e c t o r b u n d l e s a n d h o m o g e n e o u s s p a c e s , P r o c . Sympos. P u r e M a t h . , v o l . 3, A M S , 1961, pp. 7-38. [AH62] M . A t i y a h and F . H i r z e b r u c h , A n a l y t i c c y c l e s o n c o m p l e x m a n i f o l d s , Topology 1 (1962), 25-46. [AH75] D . R. A n d e r s o n and W . - C . H s i a n g , T h e f u n c t o r s a n d p s e u d o - i s o t o p i e s of p o l y h e d r a , A n n a l s of M a t h . 105 (1977), 201-223. [A62] [A63]

THE

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23

[ALS85] J . A i s b e t t , E . L l u i s - P u e b l a and V . S n a i t h , O n K * ( Z / n ) a n d K*(¥ [t]/(t )), M e m o i r s A M S , vol. 57, A M S , 1985. [B69] A . B a k , S t a b l e s t r u c t u r e of Q u a d r a t i c M o d u l e s , C o l u m b i a University thesis (1969). [Ba61] H . Bass, P r o j e c t i v e m o d u l e s o v e r a l g e b r a s , A n n a l s of M a t h . 73 (1961), 532-542. [Ba62] H . Bass, T o r s i o n f r e e a n d p r o j e c t i v e m o d u l e s , Trans. A M S 102 (1962), 319-327. [Ba64] H . Bass, K - t h e o r y a n d s t a b l e a l g e b r a , P u b l . I H E S 22 (1964), 1-60. [Ba72] H . Bass Some problems i n " c l a s s i c a l " algebraic K - t h e o r y , Algebraic K-theory II, Lecture q

2

Notes i n M a t h . , vol. 342, Springer Verlag, 1973, pp. 1-73. H . Bass, U n i t a r y a l g e b r a i c K - t h e o r y , Algebraic K-theory III, Lecture Notes i n M a t h . , vol. 343, Springer Verlag, 1973, pp. 57-265. [Ba75] H . Bass, A l g e b r a i c K - t h e o r y : a h i s t o r i c a l s u r v e y , P r o c . I C M Vancouver 1974, vol. I, 1975, p p . 277-283. [BaHS] H . Bass, A . Heller and R. Swan, T h e W h i t e h e a d g r o u p of a p o l y n o m i a l e x t e n s i o n , P u b l . I H E S 22 (1964), 61-79. [BaLS] H . Bass, M . L a z a r d and J . - P . Serre, S o u s - g r o u p e s d ' i n d i c e fini d a n s S L ( n , I , ) , B u l l . A M S 70 (1964), 385-392. [BaMS] H . Bass, J . M i l n o r and J . - P . Serre, S o l u t i o n of t h e c o n g r u e n c e s u b g r o u p p r o b l e m f o r S L , ( n > 3) a n d S p 2 n , (n > 2) 33 (1967), P u b l . I H E S , 59-137. [BaMur] H . Bass and M . P . M u r t h y , G r o t h e n d i e c k g r o u p s a n d P i c a r d g r o u p s of a b e l i a n g r o u p r i n g s , A n n . of M a t h . 86 (1967), 16-73. [BaRoy] H . Bass and A . R o y , L e c t u r e s o n t o p i c s i n a l g e b r a i c K - t h e o r y , Lectures on M a t h e m a t i c s , N o . 41, T a t a Inst. F u n d . Res., B o m b a y , 1967. [BaSch] H . Bass and S. Schanuel, T h e h o m o t o p y t h e o r y of p r o j e c t i v e m o d u l e s , B u l l . A m e r . M a t h . Soc. 68 (1962), 425-428. [BaT] H . Bass and J . Tate, T h e M i l n o r r i n g of a g l o b a l field, Algebraic K - t h e o r y II, Lecture Notes i n M a t h . , vol. 342, Springer Verlag, 1973, pp. 349-446. [Bass] H . Bass, A l g e b r a i c K - t h e o r y , B e n j a m i n , 1968. [Bei80] A . B e i l i n s o n , H i g h e r r e g u l a t o r s a n d v a l u e s of L - f u n c t i o n s of c u r v e s , F u n k t . A n a l , i P r i l o z h e n . 14 (1980), 46-47 (Russian); F u n c t i o n a l A n a l . A p p l . 14 (1980), 116-118. [Bei82] A . B e i l i n s o n , L e t t r e d e B e i l i n s o n a S o u l e , (Jan. 11, 1982). [Bei84] A . B e i l i n s o n , H i g h e r r e g u l a t o r s a n d v a l u e s of L - f u n c t i o n s , Itogi N a u k i i T e k h n i k i 24 (1984), 181-238 (Russian); J . Soviet M a t h e m a t i c s 30 (1985), 2036-2070. [BG73] K . B r o w n and S. Gersten, A l g e b r a i c K - t h e o r y a s g e n e r a l i z e d sheaf c o h o m o l o g y , Algebraic K - t h e o r y I, Lecture Notes i n M a t h . , vol. 341, Springer-Verlag, 1973, pp. 266-292. [Bir69] B. J . Birch, K of g l o b a l fields, P r o c . S y m p . P u r e M a t h . , vol. X X , 1971, pp. 87-95. [BHM] M . Bokstedt, W . - C . H s i a n g and I. M a d s e n , T h e c y c l o t o m i c t r a c e a n d t h e K - t h e o r e t i c a n a l o g u e of N o v i k o v ' s c o n j e c t u r e , P r o c . N a t . A c a d . Sci. U . S . A . 86 (1989), 8607-8609; T h e c y c l o t o m i c t r a c e a n d a l g e b r a i c K - t h e o r y of s p a c e s , Inv. M a t h . I l l (1993), 465-539. [B174] S. B l o c h , K of a l g e b r a i c c y c l e s , A n n a l s of M a t h . 99 (1974), 349-379. [B177] S. B l o c h , H i g h e r r e g u l a t o r s , a l g e b r a i c K - t h e o r y , a n d z e t a f u n c t i o n s of e l l i p t i c c u r v e s , unpublished lecture notes (1977), U n i v . C a l i f o r n i a at Irvine; A l g e b r a i c K - t h e o r y a n d z e t a f u n c t i o n s of e l l i p t i c c u r v e s , P r o c . I C M H e l s i n k i 1978, 1980, pp. 511-515. [B178] S. B l o c h , A l g e b r a i c K - t h e o r y a n d c r y s t a l l i n e c o h o m o l o g y , P u b l . I H E S 47 (1978), 187¬ 268. [B179] S. B l o c h , T o r s i o n a l g e b r a i c c y c l e s a n d a t h e o r e m of R o i t m a n , C o m p o s . M a t h . 39 (1979), 107-127. [B180] S. B l o c h , L e c t u r e s o n A l g e b r a i c C y c l e s , D u k e U n i v . M a t h . Dept., 1980. [Ba73]

n

2

2

[BlOgs] [Bo72]

[Bo77] [B685]

S. B l o c h and A . Ogus, G e r s t e n ' s c o n j e c t u r e a n d t h e h o m o l o g y of s c h e m e s , A n n . Sc. N o r m . 7 (1974), 181-202. A . Borel, C o h o m o l o g i e r e e l e s t a b l e d e g r o u p e s S - a r i t h m e t i q u e s c l a s s i q u e s , C.R. Sci. P a r i s 274 (1972), A 1 7 0 0 - A 1 7 0 2 ; S t a b l e real c o h o m o l o g y of a r i t h m e t i c g r o u p s , Sci. Ecole N o r m . Sup. 7 (1974), 235-272. A . Borel, C o h o m o l o g i e d e S L et valeurs de fonctions zeta a u x points entiers, Scuola N o r m . Sup. P i s a 4 y r l 9 7 7 , 613-636; E r r a t a 7 (1980), 373. M . Bokstedt, T o p o l o g i c a l H o c h s c h i l d h o m o l o g y , preprint 1985 (unpublished). m

Ecole Acad. Ann. Ann.

24

C H A R L E S A. W E I B E L

A . B o r e l and J . - P . Serre, L e t h e o r e m e d e R i e m a n n - R o c h , B u l l . Soc. M a t h . France 86 (1958), 97-136. [Br28] R. B r a u e r , U n t e r s u c h u n g e n i i b e r d i e a r i t h m e t i s c h e n E i g e n s c h a f t e n v o n G r u p p e n l i n e a r e r S u b s t i t u t i o n e n . I . , M a t h . Z. 28 (1928), 677-696. [Br73] K . B r o w n , A b s t r a c t h o m o t o p y t h e o r y a n d g e n e r a l i z e d sheaf c o h o m o l o g y , Trans. A M S 186 (1973), 419-458. [Br76] W . Browder, A l g e b r a i c K - t h e o r y w i t h c o e f f i c i e n t s Z / p , Lecture Notes i n M a t h . , vol. 657, Springer-Verlag, 1978, pp. 40-84. [ C C G H ] R. Cohen, G . C a r l s s o n , T . G o o d w i l l i e and W . - C . H s i a n g , T h e f r e e l o o p s p a c e a n d t h e a l g e b r a i c K - t h e o r y of s p a c e s , K - t h e o r y 1 (1987), 53-82. [СЕ] H . C a r t a n and S. Eilenberg, H o m o l o g i c a l A l g e b r a , P r i n c e t o n U. Press, 1956. [Ch36] C . Chevalley, L ' A r i t h m e t i q u e d a n s l e s a l g e b r e s d e m a t r i c e s , A c t u a l i t e s Sci. et Ind. 232 (1936), H e r m a n n , P a r i s . [CSS82] J . - L . Colliot-Thelene, J . - J . Sansuc and C . Soule, Q u e l q u e s t h e o r e m e s d e finitude en t h e o r i e d e s c y c l e s a l g e b r i q u e s , C . R. A c a d . Sci. P a r i s Ser. I M a t h . 294 (1982), 749-752. [D43] J . Dieudonne, L e s d e t e r m i n a n t s s u r u n c o r p s поп c o m m u t a t i f , B u l l . Soc. M a t h . France 71 (1943), 27-45. [D76] R. K . Dennis, H i g h e r a l g e b r a i c K - t h e o r y a n d H o c h s c h i l d h o m o l o g y , unpublished preprint (1976). [DF82] W . D w y e r and E . Friedlander, E t a l e K - t h e o r y a n d a r i t h m e t i c , B u l l . A M S 6 (1982), 453-455. [ D F S T ] W . Dwyer, E . Friedlander, V . S n a i t h and R . W . T h o m a s o n , A l g e b r a i c K - t h e o r y e v e n t u ­ a l l y s u r j e c t s o n t o t o p o l o g i c a l K - t h e o r y , Invent. M a t h . 66 (1982), 481-491. [DHS80] W . Dwyer, W . - C . H s i a n g and R. Staffeldt, P s e u d o - i s o t o p y a n d i n v a r i a n t t h e o r y , T o p o l ­ ogy 19 (1980), 367-385. [DI82] R. K . Dennis and K . Igusa, H o c h s c h i l d h o m o l o g y a n d t h e s e c o n d o b s t r u c t i o n f o r p s e u d o i s o t o p y , Lecture Notes i n M a t h . , vol. 966, Springer Verlag, 1982, pp. 7-58. [DM94] B . D u n d a s and R. M c C a r t h y , S t a b l e K - t h e o r y a n d t o p o l o g i c a l H o c h s c h i l d h o m o l o g y , A n n a l s of M a t h . 140 (1994), 685-701. [DS73] R. K . Dennis and M . Stein, K - i of r a d i c a l i d e a l s a n d s e m i - l o c a l r i n g s r e v i s i t e d , Algebraic K - t h e o r y II, Lecture Notes i n M a t h . , vol. 342, Springer Verlag, 1973, pp. 281-303. [DS75] R. K . Dennis and M . Stein, K - i of d i s c r e t e v a l u a t i o n r i n g s , Advances i n M a t h . 18 (1975), 182-238. [DW] R. Dedekind and M . Weber, T h e o r i e d e r a l g e b r a i s c h e n F u n k t i o n e n e i n e r V e r a n d e r l i c h e n , (= R. D e d e k i n d , G e s a m . M a t h . Werke vol. I, Braunschweig, (1932), pp. 238-349, Crelle's J o u r n a l 92 (1882), 181-290. [EF80] L . Evens and E . Friedlander, K ( Z / p ) a n d K ( Z / p [ e ] ) f o r p > 5 a n d r < 4, B u l l . A M S 2 (1980), 440-443. [FH70] F . T . Farrell and W . - C . H s i a n g , A f o r m u l a f o r K i R [ T ] , P r o c . S y m p . P u r e M a t h . , vol. 17, A M S , 1970, pp. 192-218. [FH76] F . T . F a r r e l l and W . - C . H s i a n g , O n t h e r a t i o n a l h o m o t o p y g r o u p s of t h e d i f f e o m o r p h i s m g r o u p s of d i s c s , s p h e r e s a n d a s p h e r i c a l m a n i f o l d s , P r o c . S y m p . P u r e M a t h . , vol. 32, A M S , 1978, p p . 325-337. [Fo69] R. Fossum, V e c t o r b u n d l e s o v e r s p h e r e s a r e a l g e b r a i c , Invent. M a t h . 8 (1969222-225). [Gar71] H . G a r l a n d , A finiteness t h e o r e m f o r K - i of a n u m b e r field, A n n a l s of M a t h . 94 (1971), 534-548. [Ger69] S. Gersten, O n t h e f u n c t o r K , J . A l g e b r a 17 (1971), 212-237. [Ger70] S. Gersten, O n M a y e r - V i e t o r i s f u n c t o r s a n d a l g e b r a i c K - t h e o r y , J . A l g e b r a 18 (1971), 51-88. [Ger72] S. Gersten, H i g h e r K - t h e o r y of r i n g s , Algebraic K - t h e o r y I, Lecture Notes i n M a t h . , vol. 341, Springer Verlag, 1973, pp. 3-43; S o m e e x a c t s e q u e n c e s i n t h e h i g h e r K - t h e o r y of r i n g s , i b i d . , pp. 211-243. [Ger73] S. Gersten, H i g h e r K - t h e o r y f o r r e g u l a r s c h e m e s , B u l l . A M S 79 (1973), 193-195. [Ger74] S. Gersten, K - t h e o r y of f r e e r i n g s , C o m m . A l g e b r a 1 (1974), 39-64. [Gi80] H . G i l l e t , R i e m a n n - R o c h t h e o r e m s f o r h i g h e r a l g e b r a i c K - t h e o r y , B u l l . A M S 3 (1980), 849-852; Advances i n M a t h . 40 (1981), 203-289. [BoSe]

r

2

r

a

2

THE [Glet] [Golo] [GQ76] [GQ82] [Gr57]

[Gr58] [Gr77] [GW83] [Gw86] [GWL]

[H40] [H64] [H65] [HR64] [HS75] [HS82] [HW73] [J73] [K52] [K58] [K59] [K80]

[K82] [Ka68]

[Ka73] [Kas80] [Keu71]

[Keu78] [Keu80]

D E V E L O P M E N T OF ALGEBRAIC if-THEORY

25

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n

n

n

2

[MS82]

[MS90]

[Mu65] [Mu66] [NV68] [088]

A . Merkurjev a n d A . Suslin, K - c o h o m o l o g y of S e v e r i - B r a u e r v a r i e t i e s a n d t h e n o r m r e s i d u e h o m o m o r p h i s m , Izv. A k a d . N a u k С С С Р 46 (1982) (Russian); M a t h . U S S R Izvestiya 21 (1983), 307-340. A . M e r k u r j e v a n d A . Suslin, T h e n o r m r e s i d u e h o m o m o r p h i s m of d e g r e e t h r e e , Izv. A k a d . N a u k С С С Р 54 (1990), 339-356 (Russian); M a t h . U S S R Izvestiya 36 (1991), 349-367. M . P . M u r t h y , P r o j e c t i v e m o d u l e s o v e r a c l a s s of p o l y n o m i a l r i n g s , M a t h . Zeit. 88 (1965), 184-189. M . P . M u r t h y , P r o j e c t i v e A [ X ] - m o d u l e s , J . L o n d o n M a t h . 41 (1966), 453-456. A . Nobile and M . K a r o u b i , S u r l a K - t h e o r i e a l g e b r i q u e , A n n . S c i . E c . N o r m . S u p . (Paris) 1 (1968), 581-616. R. Oliver, W h i t e h e a d G r o u p s of F i n i t e G r o u p s , Cambridge U n i v . Press, 1988.

THE [PS] [Q70]

[Q71] [Q72]

[Q73] [Q73Z] [Q75] [Q76] [Q94] [Q4-] [Qlet] [R57] [R59] [R61] [R71] [Rob76] [Rosen] [RW97] [Se55] [Se58] [Seg74] [SGA6] [Sh78] [Shuk] [Sie65]

[Sou79] [Sou80]

[Ssh58] [St62]

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S h e r m a n , T h e K - t h e o r y of a n e q u i c h a r a c t e r i s t i c d i s c r e t e v a l u a t i o n r i n g i n j e c t s i n t o t h e K - t h e o r y of i t s field of q u o t i e n t s , Pacific J . M a t h . 74 (1978), 497-499. U . Shukla, C o h o m o l o g i e d e s a l g e b r e s a s s o c i a t i v e s , A n n . Sci. E c . N o r m . Sup. 78 (1961), 163-209. L . Siebenmann, O b s t r u c t i o n t o finding a b o u n d a r y f o r a n o p e n m a n i f o l d of d i m e n s i o n g r e a t e r t h a n five, P r i n c e t o n U n i v e r s i t y thesis (1965); T h e s t r u c t u r e of t a m e e n d s , Notices A M S 13 (1966), 862. C . Soule, K - t h e o r i e d e s оппеомж d ' e n t i e r s d e c o r p s d e n o m b r e s e t c o h o m o l o g i e e t a l e , Invent. M a t h . 55 (1979), 251-295. C . 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