The development of Algebraic K-theory before 1980

Charles A. Weibel

Dedicated to Hyman Bass

Algebraic K-theory had its origins in many places, and it is a matter of taste to decide whether or not early papers were K-theory. We shall think of early results as pre-history, and only mention a few typical examples. One early example is the introduction in 1845 of Grassmann varieties by Cayley and Grassmann. This formed the core of the classifying space notions in K-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 Witt 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 in 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-theory [Mag] was used frequently in preparing this article. Various other survey articles were very useful to us, including [Mi66, Sw70, Ba75, Lo76, Va76, O88]. 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 in geometric topology and one in algebraic geometry. The first was Whitehead’s construction of the Whitehead group W h(π) of a group π as an obstruction in 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 K-theory. Suppose given a homotopy

\begin{matrix}c \\ ◯ \end{matrix}1999 American Mathematical Society

The author was supported in part by NSF Grants.

equivalence f between two complexes (simplicial in 1939 and 1941, cellular in 1950). Whitehead found an obstruction, called the torsion τ(f), for f to be built up out of “simple” moves (expansions and contractions). Unlike the Reidemeister torsion of f, which is a real number, Whitehead constructed τ(f) as an element of a universal group, now called the Whitehead group W h(π), which depends only upon the fundamental group π. The thesis of Whitehead’s student G. Higman [H40] contained some calculations of elements of W h(π) coming from units in ℤπ. However, the group W h(π) proved to be intractable, and the subject was put aside until 1961.

In 1961, Milnor used Reidemeister torsion in [Mi61] to disprove the Hauptvermutung, the conjecture that two polyhedral decompositions of a space always have a common subdivision. The Hauptvermutung dated back to Poincaré’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 Poincaré’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 [Mi61] Milnor coined the term Whitehead group for the quotient GL(R)/E(R) of the general linear group GL(R) by the subgroup E(R) generated by the elementary matrices — the group we now call K_{1}(R). Milnor also clarified that Whitehead’s torsion invariant lies in the quotient of this group by ±π. This quotient would not be called the Whitehead group W h(π) until [Mi66], but let us get back to the story.

The second origin of K-theory came from algebraic geometry, and was immediately influential. It was the key ingredient in Grothendieck’s 1957 reformulation of the Riemann-Roch theorem [BoSe] [Gr57]. Here Grothendieck introduced a group K(A) associated to a subcategory A of an abelian category. This was the real beginning of K-theory; Grothendieck chose the letter ‘K’ for “Klassen” (the German word for classes); today we write K_{0}(A) for this group and call it the Grothendieck group of A.

Grothendieck was interested in the abelian category M(X) of coherent sheaves on an algebraic variety X, together with its subcategory P(X) of locally free sheaves on X. He defined what we now call K_{0}(X) and G_{0}(X) as K_{0}P(X) and K_{0}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

G_{0}(Y) → G_{0}(X) → G_{0}(X − Y) → 0.

Grothendieck also defined the γ-filtration on K_{0}(X), and showed that the associated graded ring is isomorphic (up to torsion) to the Chow group A^{∗}(X), via Chern classes c_{i}: K_{0}(X) → A^{i}(X) [Gr58]. Moreover, the total Chern character induces a ring isomorphism ch_{X}: K_{0}(X) ⊗ ℚ ≅ A^{∗}(X) ⊗ ℚ. If f: Y → X is proper he showed that the formula

[F] ↦ ∑(−1)^{i}[R^{i}f_{∗}(F)]

THE DEVELOPMENT OF ALGEBRAIC K-THEORY 3

defines a covariant map f_{∗}: G_{0}(Y) → G_{0}(X), and used this to formulate the Grothendieck Riemann-Roch theorem: if X and Y are nonsingular and T_{X} is the Todd class of X then in A^{∗}(X) ⊗ ℚ

ch_{X}(f_{∗}(y)) · T_{X} = f_{∗}(ch_{Y} (y) · T_{Y}), y ∈ K_{0}(Y).

The formula G_{0}(X) ≅ G_{0}(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[t]). For nonsingular X this meant that K_{0}(X) ≅ K_{0}(X[t]). In particular, for X = Spec(k) this implies that K_{0}(k[x_{1}, ..., x_{n}]) ≅ K_{0}(k) ≅ ℤ. In other words, every projective module P over R = k[x_{1}, ..., x_{n}] satisfies P ⊕ R^{m} ≅ R^{n} for some m and n.

The rise of topological K-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_{i} (combined into the Chern character ch) defines a ring homomorphism ch: K(X) → 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^{−n}(X, Y) = K̃(S^{n}(X/Y)), 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 periodicity” isomorphism K^{n}(X) ≅ K^{n+2}(X), they extended K^{n}(X, Y) to all n using periodicity. Then they showed that the functors K^{n}(X, Y) form an extraordinary cohomology theory, satisfying all the Eilenberg-Steenrod axioms except the dimension axiom. They were able to calculate the K-theory of certain homogeneous spaces using what we now call the Atiyah-Hirzebruch spectral sequence:

E_{2}^{pq} = H^{p}(X, K^{q}(point)) ⟹ K^{p+q}(X).

What made these new constructions so appealing was the power they brought to applications. We have already mentioned the Riemann-Roch theorem for differentiable manifolds above [AH59a]. In [AH59b] they were able to show that complex projective space ℂℙ^{n} cannot be embedded in ℝ^{N} for N = 4n − 2α(n), where α(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 u ∈ H^{2k}(X; ℤ) which is represented by a complex analytic subvariety must satisfy Sq^{3}u = 0. This allowed them to disprove an overoptimistic conjecture of Hodge.

Ring homomorphisms ψ^{k}: K(X) → K(X), which we now call Adams operations, 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^{n}, which had been a famous outstanding problem. By further analyzing the effect of the ψ^{k} on the rings K(X), Adams was able to describe the image of the Hopf-Whitehead J-homomorphism J: π_{i}(SO_{n}) → π_{n+i}(S^{n}) in terms of the denominator of the Bernoulli numbers B_{k}/4k. His ideas were based upon [AH59a], and discovered in 1961/1962, but his proof was not completely published until 1966 [A63].

At this point, the subject was off and running. But after 1962, the development of topological K-theory diverged somewhat from the development of algebraic Ktheory. 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 circa1953, 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 [CE]. On p. 11 we find the characterization of semisimple rings as rings for which every module is projective; later in the book (p. 111), 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 Prüfer domain), 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 R-modules are free when R is a principal ideal domain (or more generally a Bezóut 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 R-modules are free. Kaplansky later showed [K58] that all projective modules over a local ring are free, as a consequence of the general result 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 Rmodule is free when R = k[x_{1}, ..., x_{N}] is a polynomial ring over a field. This became known as the “Serre problem.” Serre’s 1957 result that K_{0}(R) ≅ ℤ meant that every projective R-module P is stably free: P ⊕ R^{m} ≅ R^{n} 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].

Serre’s 1955 paper [Se55] had a more long-reaching effect: it set up a dictionary between projective modules and topological vector bundles. The analogy was strengthened by his 1958 paper [Se58], which showed that every projective R-module has the form P ⊕ R^{n}, 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.

After 1962 people searched for projective R[t]-modules which were not extended, i.e., of the form P [t]. 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 IX of [Ba72]) that if R is a commutative regular ring then every projective R[t]-module is extended; Bass’ conjecture is still open when dim(R) ≥ 3.

The rise of algebraic K-theory (1957–1964)

We have seen that Grothendieck’s projective class group was immediately useful in geometric settings, for studying vector bundles in algebraic geometry and in topology. We now turn to the development of the Grothendieck group construction in a completely algebraic setting.

For now, let us write K(R) for the Grothendieck group of finitely generated projective modules over a ring R. If R is a commutative ring, a good description of K(R) could be read off from the results in [BoSe] for the scheme X = Spec(R): K(R) is a commutative ring equipped with operations λ^{k}, and its associated γfiltration is related to the Krull dimension of R.

The group K(R) next appeared in the 1958 paper [R59] by Dock Rim. Writing Γ[R] for the reduced group K(R)/ℤ, Rim first restated an old result of Chevalley [Ch36] as saying that if R is Dedekind then Γ[R] is the ideal class group of R. Then he considered the group ring ℤπ of a finite cyclic group π of order p. Combining his results on projective ℤπ-modules with those of Reiner [R57], Rim then showed that K(ℤπ) ≅ K(ℤ[ζ]), where ζ is a primitive p-th root of unity.

Next came Swan’s 1959 paper [Sw59], whose details appeared in [Sw60]. Here Grothendieck’s methods were used in a deep way. In order to classify the (fin. gen.) projective modules over the group ring ℤπ of a finite group π, Swan considered the Grothendieck group K(Rπ) of projective Rπ-modules (which he called P (Rπ)). Noting the analogy between this and the representation ring of π, Swan used induction techniques to prove that the group K(Rπ) injects into the sum of the groups K(Rπ′) as π′ ranges over all hyperelementary subgroups of π. To do this, he introduced the Grothendieck groups G(Rπ) and G′(Rπ), corresponding respectively to the classes of all (finitely generated) Rπ-modules, and all those which are torsionfree. If R is Dedekind, these Grothendieck groups are isomorphic.

Rim’s second paper [R61] contained the germ of the Mayer-Vietoris sequence. Let B be a finite ring extension of a ring A, and consider the conductor ideal I = ann(B/A). In the special case when A is an order in a semisimple algebra (such as a group ring), and B is a maximal order, Rim showed that there is an exact sequence

Γ[A] → Γ[B] ⊕ Γ[A/I] → Γ[B/I] → 0.

In retrospect, Rim was unable to extend this result on the left (this was done in [BaMur]) because K_{1} was not yet available; Rim had no extension on the right because A was one-dimensional. (The extension to the right uses K_{−1} and was done in [Bass, p. 677].)

Up to this point the Grothendieck group K(R) had not become K_{0}(R), because there was no analogue K_{1}(R) of the topologist’s K^{−1}(X). This changed with the short 1962 announcement [BaSch] by Bass and Schanuel; the details appeared in the expanded version, published by Bass in [Ba64]. Motivated by Serre’s dictionary with vector bundles, and the observation that the topologist’s formula K^{1}(X) = [X, U] has an algebraic sense, they wrote K^{0}(R) for K(R) and defined the group K^{1}(R) to be the quotient GL(R)/E(R). Like Milnor in [Mi61], they called K^{1} the Whitehead group, explaining that they were “using notions visibly borrowed from Whitehead’s theory of simple homotopy types” [Wh50]; these “notions” included Whitehead’s theorem that E(R) is the commutator subgroup of GL(R).

They also introduced the stability problem: for which n are GL_{n}(R)/E_{n}(R) isomorphic to K_{1}(R)? Dieudonné’s theory [D43] of noncommutative determinants implied that n = 1 suffices for a division ring D: K_{1}(D) = D^{×}/[D^{×}, D^{×}]. Bass introduced his stable range conditions in [Ba64, p. 14] in order to solve this problem. Looking forward, Bass’ stable range results were improved in [Bass] and again in Vaserstein’s work [Va69] [Va71]. A later milepost in the stability problem was Suslin’s result [Su77] that SL_{3}(F [x_{1}, ..., x_{n}]) is generated by elementary matrices.

If f: R → S is a ring homomorphism, Bass and Schanuel also defined a relative group K^{0}(f) fitting into an exact sequence

K^{1}(R) → K^{1}(S) → K^{0}(f) → K^{0}(R) → K^{0}(S).

Bass also defined the relative group K^{1}(R, I) = GL(R, I)/E(R, I) of an ideal I in [Ba64], showing that it extended the above sequence for f: R → R/I by one term. This paper also defined products such as K^{0}(R_{1}) ⊗ K^{1}(R_{2}) → K^{1}(R_{1} ⊗ R_{2}).

One important application of these results was given in [Ba64] (and announced in [BaSch]): if R is finitely generated as a module over ℤ then both K^{0}(R) and K^{1}(R) are finitely generated abelian groups. Consequently GL_{n}(R) and GL_{n}(R, I) are finitely generated groups for all n.

Bass, Heller and Swan published the companion paper [BaHS] in the same issue as [Ba64]. In it, they systematically studied the behavior of the new functor K^{1}(R) under polynomial and related extensions. If R is left regular, they proved that K^{1}(R) = K^{1}(R[t]) and K^{1}(R[t, t^{−1}]) ≅ K^{1}(R) ⊕ K^{0}(R). The map K^{1}(R[t, t^{−1}]) → K^{0}(R) was constructed for any ring R; its inverse is multiplication by t ∈ K^{1}(R[t, t^{−1}]).

A key step in their proof [BaHS, p. 551] was the introduction of a group K^{1}(A) for any subcategory A of an abelian category, along with a devissage theorem for K^{1}. Alex Heller abstracted this in [H65] to introduce groups K_{1}(S) and K_{0}(S) for any symmetric monoidal category S. Bass then systematically developed Heller’s ideas in his Tata Lecture notes [BaRoy].

In 1964, a consensus arose that one should write K_{0}(R) and K_{1}(R) instead of K^{0}(R) and K^{1}(R), to indicate that these new functors were covariant in the ring R. I believe that the first uses of this new notation were in two papers written in 1964: Heller’s paper [H65] and Milnor’s survey paper [Mi66] on Whitehead Torsion. Milnor’s paper also introduced the notation K̃_{0}(R) for Rim’s group Γ[R].

Motivated by the classical unit/ideal class group sequences for Dedekind rings, Alex Heller turned to the construction of a localization sequence for K_{i}(ℤπ) → K_{i}(ℚπ). His paper [HR64] with Reiner had partial success, using the group K_{0}B of torsion ℤπ-modules. In [H65], Heller extended Grothendieck’s K_{0} localization sequence [Gr57] for a Serre subcategory B of an abelian category A to an exact sequence (at least when A and A/B are semisimple)

K_{1}A → K_{1}A/B → K_{0}B → K_{0}A → K_{0}A/B → 0.

It became apparent that the Bass-Heller-Swan group K_{1}A was not the appropriate group when A is abelian but not semisimple. This was confirmed by Leslie Roberts, whose 1968 thesis ([Rob76] [Ger72, 5.2]) showed that Bass’ group K^{1}(A) doesn’t fit into a localization sequence with K^{1}(B) when A is the category of vector bundles over an elliptic curve. But the way to repair K_{1}A did not become clear until 1972.

The foundations for K_{0} and K_{1} were now in place.

Applications to Topology (1963–1966)

The next few years were marked by a veritable explosion of applications of the new K-theory. We have already pointed out the use of the Whitehead group W h(π) = K_{1}(ℤπ)/±π as an obstruction in simple homotopy theory since 1950, and its subsequent use by Milnor [Mi61] in disproving the Hauptvermutung in 1961.

Cobordisms also formed an important family of applications of W h(π). Recall that an h-cobordism (W, M, M′) is a smooth closed manifold W whose boundary is the disjoint sum of two pieces, M and M′, each of which are deformation retracts of W. If M is simply connected and dim(M) ≥ 5, Smale proved the h-cobordism theorem in 1962: W is diffeomorphic to the product M × I (in a way preserving M), so that M′ is diffeomorphic to M. The first part of the following generalization was proven in 1963 by Barden, Mazur and Stallings; see [Mi66]. The second part was proven by J. Stallings [St65].

The s-cobordism Theorem. If dim(M) ≥ 5, then an h-cobordism (W, M, M′) is diffeomorphic to the product M×I if and only if the Whitehead torsion τ(W, M) ∈ W h(π_{1}M) vanishes.

Moreover, Whitehead torsion gives a 1-1 correspondence between the elements of W h(π_{1}M) and diffeomorphism classes of h-cobordisms (W, M, M′) of M.

If π is finite, Bass had proven in [Ba64] that W h(π) is finitely generated, and computed its rank. It was implicit in Higman’s paper [H40] that the torsion in W h(π) comes from SL_{2}(ℤπ). (Higman worked with abelian π; the general result came much later [W74].) In spite of a heroic effort by several people in the mid-1960’s, examples of nonzero torsion proved to be elusive (see [Mi66, p. 416]), and not actually confirmed until the 1973 paper [ADS] by Alperin, Dennis and Stein. We refer the reader to Oliver’s book [O88] for the rest of this intricate story.

An important application of K_{0} was found by C. T. C. Wall in 1963, and published in [W65]. If X is a space dominated by a finite complex, he defined a generalized Euler characteristic χ(X) in K̃_{0}(ℤπ). Wall showed that X is homotopy equivalent to a finite complex if and only if χ(X) = 0. Naturally, we now call χ(X) Wall’s finiteness obstruction.

L. Siebenmann [Sie66] and V. Golo [Golo] soon showed that similar obstructions exist in K̃_{0}(ℤπ) to the problem of putting a boundary on an open manifold.

One of the most important K-theory problems in this era was to compute the group K̃_{0}(ℤπ) containing these obstructions. When π is abelian, this was accomplished by Bass and Murthy in [BaMur]. Their idea was to decompose π as the product of a finite group π_{0} and a free abelian group T and then write ℤπ = A[T] with A = ℤπ_{0}. If B is the normalization of A, this could then be analyzed by extending Rim’s conductor square sequence to K_{1} and studying the units of B/I[T].

When π = π_{0} ⋊ ℤ is a semidirect product with ℤ, Farrell and Hsiang [FH70] found a general formula for K_{1}ℤπ and W h(π), generalizing the Bass-Murthy formula.

The Congruence Subgroup Problem

The other important application of K-theory in the mid-1960’s was the congruence subgroup problem. Let R be the ring of integers in a global field F. If I is an ideal of R, let SL_{n}(I) denote the kernel of SL_{n}(R) → SL_{n}(R/I). Since R/I is a finite ring, SL_{n}(I) has finite index in SL_{n}(R). The congruence subgroup problem asks if every subgroup Γ of finite index contains a subgroup SL_{n}(I) for some I, so that it corresponds to a subgroup of the finite group SL_{n}(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 SL_{2}(ℤ).

If n ≥ 3, we see K-theory appearing: the answer to the congruence subgroup problem for SL_{n}(R) is ‘yes’ if and only if SK_{1}(R, I) = 0 for every nonzero ideal I of R. Indeed, for any nonzero I the group E_{n}(R, I) also has finite index, and Bass’ stability theorems in [Ba64] show that the quotient SL_{n}(I)/E_{n}(R, I) is isomorphic to the subgroup SK_{1}(R, I) of K_{1}(R, I). Conversely, if Γ ⊂ SL_{n}(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_{n}(R, I) ⊆ Γ ⊆ SL_{n}(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

subgroup SL_{2}(R) of SL_{3}(R), modulo E_{3}(R). The class of a 2 × 2 matrix ( \begin{matrix}a & b \\ c & d \end{matrix} )

only depends upon the first row (a b) of the matrix, and Mennicke showed that this class is multiplicative in a and b. This led Milnor to introduce the universal Mennicke group generated by symbols [{b \atop a}], where b ∈ I and a ∈ 1 + I are relatively prime, subject to Mennicke’s two relations; this universal group maps to SK_{1}(R, I) by sending the Mennicke symbol [{b \atop a}] to the class of a 2 × 2 matrix with first row (a b). Milnor observed that this could be used to bound SK_{1}(R) when R is a ring of algebraic integers. Then Kubota used power residue symbols to produce characters on congruence subgroups (in the totally imaginary setting). Bass generalized Kubota’s arguments, using stability theorems for K_{1}, to prove that the universal Mennicke group is isomorphic to SK_{1}(R, I) for every Dedekind ring R.

The congruence problem was finally settled for all global fields in the 1967 paper [BaMS] by Bass, Milnor and Serre. The answer is ‘yes’ if and only if F has an embedding into ℝ. If there is no such embedding, then for each ideal I they give a formula for a number m so that the group μ_{m} of mth roots of unity lie in F and SK_{1}(R, I) ≅ μ_{m}. Under this isomorphism, the Mennicke symbol [{b \atop a}] corresponds to Hasse’s classical power residue symbol (\frac{b}{a})_{m}.

The solution revealed an unexpected interaction between the relative groups K_{1}(R, I) and explicit power reciprocity laws in F. This was to attract the attention of many people to K-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_{2}, and form the backdrop for the appearance of the higher K-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 K_{0}(S) and K_{1}(S) was systematically exposed. On the next page, we shall say more about the applications to S = Az 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 interpretation of the localization sequence for A → S^{−1}A using the category M_{S}(A) of S-torsion A-modules. For an integral extension A → B with conductor ideal I, they showed that excision holds for K_{0} in the sense that K_{0}(A, I) = K_{0}(B, I). This allowed them to extend Rim’s Mayer-Vietoris sequence back as far as K_{1}(A). They also proved that K_{1}(A, I) ≅ K_{1}(B, I) when B is a finite product, each factor of which is a quotient of A.

Swan’s 1968 book [Swan], based on a Fall 1967 Chicago course, gave a very readable account of the theory, focussing on K_{0} and K_{1} 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 K-theory to the representation theory of finite groups and orders, including the new induction techniques from Lam’s thesis [L68].

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_{1} of a Dedekind ring and reciprocity laws. But most importantly, it contained a proof of the “fundamental theorem” for K_{1}: the Bass-Heller-Swan map ∂ and its splitting fit into a split exact sequence:

0 → K_{1}(A) → K_{1}(A[t]) ⊕ K_{1}(A[t^{−1}]) → K_{1}(A[t, t^{−1}]) ⟶^{∂} K_{0}(A) → 0.

This sequence first appeared in [BaMur, 10.2], except that exactness at the second term was not established there. Bass and Murthy gave a similar formula for K̃_{0}(A) = Pic(A) when A is 1-dimensional. 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 K_{0} as a function LF of the functor F = K_{1}. Bass recognized that this process could be iterated to define negative K-groups K_{−n}(R) as L^{n}K_{0}(R) [Bass, p. 677]. To axiomatize the process, Bass introduced the notion of a contracted functor; the fact that K_{1} is contracted implies that K_{0} is also contracted. Independently, Karoubi [Ka68] gave a definition of K-groups K^{n}A (n ≥ 0) for any (idempotent complete) additive category A, and proved that his K^{n}P(R) agree with Bass’ K_{−n}(R).

Quadratic Forms and L-theory

The study of quadratic forms has a long and venerable history, going back to Gauss, and focussed on the Witt group W (k) of a field k, introduced by E. Witt in [Wi37]. Classical invariants include the discriminant (in k^{×}/k^{×2}) and the Hasse invariant, lying in the Brauer group Br(k). In the mid 1960’s, the idea of replacing GL_{n} by the orthogonal and unitary groups led to the “hermitian” K-theory of hermitian forms (and/or quadratic forms), and this evolved into L-theory.

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 BW (k). This was generalized to any commutative ring R in Bass’ Tata lectures [BaRoy]. Bass first formalized the notion of the category Quad(R) of quadratic forms over R and their isometries, and considered the groups K_{i}Quad(R). He observed that the Witt ring W (R) of quadratic forms is the quotient of K_{0}Quad(R) by the image of a hyperbolic map K_{0}(R) → K_{0}Quad(R). Bass also showed that K_{1}Quad(R) is related to the stable structure of the orthogonal groups O_{2n}(R).

Next, Bass introduced the category Az(R) of Azumaya algebras over a commutative ring R, and studied K_{i}Az(R). The Brauer group Br(R), defined by M. Auslander and O. Goldman in 1960, is a natural quotient of K_{0}Az(R). Bass defined the Brauer-Wall group BW (R) as the corresponding quotient of K_{0}Az_{2}(R), where Az_{2} is the graded analogue of Az. Finally, he showed that the Clifford algebra functor Quad(R) → Az_{2} induces Wall’s invariant W (R) → BW (R).

The topological part of the story comes from surgery theory. Suppose M is a closed manifold with fundamental group π. The surgery problem asks when a map ψ: 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 K̃^{0}(M) to K̃^{0}(X).

In order to solve this problem when π is non-trivial (and X is nice), C. T. C. Wall [W66] observed that the middle homotopy group of ψ is a projective ℤπ-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_{m}(π), or L_{m}(ℤπ). Wall then proved that the obstruction to the surgery problem for ψ is the class of Q in L_{m}(π), where m = dim(M).

Motivated by the topological applications, the subject rapidly evolved. Stability theorems were given by Bak [B69] and Bass’ foundational paper [Ba73], which discussed K_{0}Quad^{ϵ}(R) and K_{1}Quad^{ϵ}(R) for a ring with involution R and ϵ ∈ R such that ϵϵ̄ = 1. It turned out, however, that Wall’s 4-periodic obstruction groups L_{m}^{ϵ}(R) are subquotients of these groups [W73, p. 286], with L_{m+2}^{ϵ}(R) = L_{m}^{−ϵ}(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 K-theory was discovered, it was natural to consider the homotopy groups of the space BO^{ϵ}(R)^{+} as the Hermitian K-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_{m}^{ϵ}(R). At this point these two approaches became disconnected from each other, and also from the development of algebraic K-theory. So let us return to our story.

K_{2} arrives (1967–1971)

Steinberg [St62] gave a presentation of a universal central extension Δ of a Chevalley group G over a field k, noting that the center of Δ was a “multiplier” in the sense of Schur. When G = SL_{n}, we now write St_{n}(k) for Δ and call it the Steinberg group. Steinberg also introduced 2-cocycles k^{×} ⊗ k^{×} → St_{n}(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 St_{n}(R) of the subgroup E_{n}(R) of GL_{n}(R) for any associative ring R (if n ≥ 5).

As we have mentioned, the Bass-Milnor-Serre paper [BaMS] revealed an unexpected connection between the relative groups K_{1}(R, I) and explicit power reciprocity 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_{1}(R, I) → \frac{SL_{n}(R)}{E_{n}(R, I)} → E_{n}(R/I) → 1.

In Spring 1967, Milnor gave a course at Princeton University, in which he defined K_{2} of a ring and developed several of its basic properties. Writing St(R) for the direct limit of the groups St_{n}(R), Milnor defined K_{2}(R) to be the kernel of the homomorphism St(R) → E(R). Milnor also wrote a letter to Steinberg explaining his results on K_{2}(R). The contents of his letter were included in Steinberg’s Yale course notes on Chevalley Groups [St67, pp. 92ff.], distributed in Fall 1967.

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_{2}(F) onto the finite group μ = μ_{n} of roots of unity in F; Moore showed that its kernel V is divisible, so K_{2}(F) ≅ μ_{F} ⊕ 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_{2}(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).

Let R be a Dedekind domain with field of fractions F. For each prime ideal ℘ of R, Bass had constructed transfer maps K_{1}(R/℘) → SK_{1}(R) in [Bass, p. 451]. In 1969, Bass and Tate constructed the tame symbols λ_{℘}: K_{2}(F) → K_{1}(R/℘) and used reciprocity laws for λ = ⊕λ_{℘} similar to those in [BaMS] to construct the exact sequence

K_{2}(R) → K_{2}(F) ⟶^{λ} ⊕_{℘}K_{1}(R/℘) → SK_{1}(R) → 0

[Mi71, p. 123] [BaT]. Their proof of exactness at K_{2}(F) required R to have only countably many prime ideals, but that suffices for integers in global fields.

When R is the ring of integers in a number field F, the kernel of λ was called the tame kernel until 1972, when Quillen discovered [Q73] that the tame kernel was just K_{2}(R). In 1971, Garland [Gar71] proved that the tame kernel K_{2}(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_{2}(F) was known for these fields.

In 1970/71, J. Tate [Ta70] discovered the norm residue symbols on K_{2}(F) (one for each n with 1/n ∈ F). If F has primitive n-th roots of unity, they give maps K_{2}(F) → Br(F); in general they take values in the étale cohomology group H_{et}^{2}(F, μ_{n}^{⊗2}). Tate proved that K_{2}(F)/n → H_{et}^{2}(F, μ_{n}^{⊗2}) is an isomorphism for all global fields F, but the details were not published until several years later, in [Ta76]. The proof that this map is an isomorphism for all fields was discovered in 1981/82, by Merkurjev and Suslin [Me81] [MS82].

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

In the 1970 paper [Mi70], Milnor observed that the Hasse invariant of quadratic forms over a field F could be factored through Tate’s norm residue symbol K_{2}(F)/2 → Br(F). More precisely, if I denotes the kernel of W (F) → ℤ/2 then the Hasse invariant maps I^{2} to Br(F), and Milnor proved that it factors through an isomorphism I^{2}/I^{3} ≅ K_{2}(F)/2. This led him to introduce what we now call the Milnor K-groups of a field F, viz., a graded ring K_{∗}^{M}(F) which agrees with K_{∗}(F) for ∗ = 0, 1, 2. Milnor constructed a canonical map from K_{n}^{M}(F)/2 onto I^{n}/I^{n+1} 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 [K82].

The Milnor ring K_{∗}^{M}(F) of a field was systematically studied by Bass and Tate in [BaT]. For global fields, they were able to compute K_{n}^{M}(F) ≅ (ℤ/2)^{r_{1}} for all n ≥ 3, where r_{1} is the number of embeddings (if any) of F into ℝ. They also constructed transfer maps K_{∗}^{M}(E) → K_{∗}^{M}(F) for certain finite field extensions F ⊂ E; the proof that transfer maps exist for all finite extensions was given in 1980 by Kato [K80]. After Kato distributed his 4-page proof at the 1980 Oberwolfach conference, Suslin immediately used Kato’s transfer maps to construct a map K_{n}(F) → K_{n}^{M}(F), and prove that the natural map K_{n}^{M}(F) → K_{n}(F) is an injection modulo torsion [Su80].

Milnor also constructed higher norm residue symbols K_{n}^{M}(F)/2 ⟶^{h_{n}} H_{et}^{n}(F, ℤ/2) in [Mi70], and used Tate’s results to prove that h_{n} 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]. Jumping ahead, the case n = 3 was solved in 1986 by Merkurjev-Suslin [MS90] and M. Rost. A proof that h_{n} is an isomorphism for all n was only discovered recently by V. Voevodsky [Voe96].

For rings other than fields, progress was slower. First, the group K_{2}(R[ε])/K_{2}(R) was computed by Wilberd van der Kallen [vdK71]. Then Dennis and Stein [DS73]

[DS75] constructed symbols ⟨a, b⟩ in K_{2}(R) and showed that they form a complete set of generators for K_{2} of a semilocal ring. If I is a radical ideal, they also showed that K_{2}(R, I) is generated by the Dennis-Stein symbols ⟨a, b⟩ with a or b in I. A full set of relations for the Dennis-Stein symbols, and hence a presentation for K_{2}(R, I) was later given by Maazen and Stienstra in [MaSt].

Higher K-theory arrives (1968–1972)

The search for higher K-groups had dominated much of the 1960’s, as we have seen. It was clear that Milnor’s definition of K_{2} should form the basis of this search. In fact, Swan proposed the definition K_{3}(R) = H_{3}(St(R); ℤ) in 1967 [Swan, p.207].

In 1968, Swan [Sw68] proposed a definition of higher groups K_{n}(R), based upon the idea that free rings should have no higher K-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_{n}(R) = π_{n−2}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 K-theory, essentially by defining the “loop space” of a ring. These ideas led Karoubi and Villamayor to define groups K^{−n}(R) in [KV69] (whose details appeared in [KV71]). The drawback is that their group K^{−1}(R) is the quotient of K_{1}(R) = GL(R)/E(R) by the subgroup generated by the unipotent matrices, but it has the advantages that K^{−n}(R) ≅ K^{−n}(R[t]) for every n ≥ 1, and that long exact sequences (such as Mayer-Vietoris sequences) exist in many situations. Nowadays, we write KV_{n}(R) for K^{−n}(R) and call it the Karoubi-Villamayor K-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 GL from rings to groups. Then Rector remarked [Glet] [R71, 2.6] that to every ring R one could associate a simplicial ring ΔR with ΔR_{n} = R[t_{0}, …, t_{n}]/(∑ t_{i} = 1), and showed that the homotopy groups of the simplicial group GL(ΔR) also give the Karoubi-Villamayor groups. Rector’s remark has formed the basis of all subsequent work in the subject.

Historically, the most important construction of higher K-theory was the plus construction, given by Quillen in 1969/70. In his work [QΨ] [Q70] on the Adams Conjecture, Quillen constructed a map from BGL(𝔽_{q}) to the fiber F Ψ^{q} of the Adams operation BU ⟶^{ψ^{q}−1} BU for each finite field 𝔽_{q}, and observed that the map is acyclic (induces isomorphisms on homology). Quillen observed more generally that if one started with any CW complex X then there is an acyclic map X → X^{+}, unique up to homotopy, such that π_{1}(X) → π_{1}(X^{+}) is the quotient by the largest perfect subgroup. In particular, BGL(𝔽_{q})^{+} has the homotopy type of F Ψ^{q}.

Motivated by this, Quillen defined K_{n}(R) as the homotopy group π_{n}BGL(R)^{+} [Q71, p. 50]. Since the homotopy groups of F Ψ^{q} could be read off from the known action of ψ^{q} on BU, this gave the calculation of K_{n}(𝔽_{q}) for every finite field 𝔽_{q}. Moreover, in light of the Kahn-Priddy theorem that BΣ^{+} ≃ Ω^{∞}S^{∞}, the inclusion of the symmetric groups Σ_{n} in GL_{n}(ℤ) induces maps π_{n}^{s} → K_{n}(ℤ). Quillen showed in [Qlet] that these maps induced an injection of the cyclic groups J(π_{n}O) ⊂ π_{n}^{s} into K_{n}(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 Γ-space) and produced an Ω-spectrum. (His paper [Seg74] only appeared in 1974.) Quillen showed in the 1971 preprint (now published as [Q94]) that if one takes S to be the category of fin. gen. projective R-modules and their isomorphisms then Segal’s infinite loop space is K_{0}(R) × BGL(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 [MaTh] that all reasonable infinite loop space machines agree.

Several other constructions of K_{n}(R) appeared in 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 [Keu71].

In 1972, Quillen [Q72, Q73] defined the higher K-theory of an exact category, viz., a full subcategory A of an abelian category which is closed under extensions. To do so, he first constructed an auxiliary category QA and defined K_{n}(A) = π_{n−1}BQA. Theorem 1 of the announcement [Q72] stated that the loop space ΩBQP(R) is homotopy equivalent to K_{0}(R) × BGL(R)^{+}, so that K_{n}(R) ≅ K_{n}P(R). Awkwardly, the proof of this “+ = Q” theorem was not published until 1976 [GQ76].

Theorems 2–4 of [Q72] extended the fundamental structure theorems from classical K-theory (K_{n} for n < 2) to higher K-theory: devissage, resolution and localization. Theorem 5 stated that if R is regular then K_{∗}(R) ≅ G_{∗}(R), where G_{∗}(R) is the K-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:

G_{n}(R[t]) ≅ G_{n}(R) and G_{n}(R[t, t^{−1}]) ≅ G_{n}(R) ⊕ G_{n−1}(R).

Now Gersten had given a spectral sequence [Glet] from K_{∗}(ΔR) to the Karoubi-Villamayor groups KV_{∗}(R). If R is regular, Quillen’s Fundamental Theorem implies [Q72, p. 101] that Gersten’s spectral sequence degenerates, giving K_{∗}(R) ≅ KV_{∗}(R). Quillen concluded by defining the groups G_{∗}(X) of a quasi-projective scheme X, and giving a spectral sequence converging to G_{∗}(X), starting from the K-groups of the residue fields of X.

At this point in the rapid-fire sequence of events, Bass organized a two-week conference on algebraic K-theory. Held at the Battelle Memorial Institute in Seattle during August 28–September 8 of 1972, the relationship between the various Ktheories 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^{1}(A) differs from Quillen’s K_{1}(A) when A is the category of vector bundles over an elliptic curve.

The consensus at the Battelle Conference was that Quillen’s K-theory of a free ring ℤ{X} measures the difference between the different constructions. A few months afterward, Gersten proved [Ger74] that K_{∗}(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 in [Q72]. (We will not elaborate on the details in order to focus on the flow of events.) At that time, Quillen discovered proofs of Gersten’s conjecture and Bloch’s Formula; we will discuss these results below.

Applications to Algebraic Geometry

The interaction of higher K-theory and algebraic geometry begins with Ken Brown’s 1971 MIT thesis [Br73], which developed the cohomology theory of sheaves of spectra on a noetherian scheme X. Brown was motivated by Quillen’s suggestion that such a theory might apply to K-theory. Gersten had recently constructed a functorial spectrum 𝕂(R) associated to a ring R; sheafifying gives such a sheaf spectra. Seizing upon this, Gersten announced in [Ger73] that if we set K_{n}(X) = H^{n}(X, 𝕂) for regular separated schemes X then for X = Spec(R) we recover K_{n}(X) = KV_{n}(R). (Quillen had already announced that KV_{n}(R) = K_{n}(R) [Q72], but Gersten was unaware of this at the time.) Moreover, there is a spectral sequence converging to K_{∗}(X) whose E_{2}-terms are the Zariski cohomology H^{p}(X, K_{q}) of the sheaves associated to the presheaves sending Spec(R) to K_{q}(R). The details were worked out by Brown and Gersten at the Battelle Conference, and appeared in [BG73].

Jouanolou [J73] also extended the functors KV_{n} 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(R) → X which is locally a vector bundle; one defines KV_{∗}(X) = KV_{∗}(R). At the Battelle Conference it became clear that Quillen’s definition K_{n}(X) = K_{n}P(X) agreed, at least for regular X, with the groups obtained using Jouanalou’s device, as well as the Brown-Gersten definition [BG73, Q73].

In the other direction, the application of higher K-theory to the study of cycles in algebraic geometry was initiated by Spencer Bloch in [Bl74]. Bloch’s preprint appeared just before the 1972 Battelle Conference, and was influenced by Gersten’s results in [Ger73]. His main theorem, that the cohomology group H^{2}(X, K_{2}) is isomorphic to the Chow group CH^{2}(X) on any regular surface, is now called Bloch’s formula. One interesting conjecture made in [Bl74] was that if R is a (regular) local domain with quotient field F, then the following sequence is exact:

0 → K_{2}(R) → K_{2}(F) ⟶^{λ} ⊕_{ht℘=1}k(℘)^{×} → C_{2} → 0,

where k(℘) denotes the residue field of R_{℘}, and C_{2} 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 K-theory of regular local rings, including what we now call Gersten’s conjecture: K_{n}(R) injects into K_{n}(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^{p}(X, K_{p}) ≅ CH^{p}(X) for all p. Quillen included this material in [Q73].

Looking ahead, Gersten’s conjecture was proven for equicharacteristic DVR’s by C. Sherman in [Sh78]. The general mixed characteristic case of Gersten’s conjecture is still open today, in 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 in other settings, such as de Rham cohomology. Gillet [Gi80] later used the ideas in the Bloch-Ogus paper to define Chern classes for higher K-theory, and to prove a very general Riemann-Roch theorem.

In 1978, Bloch [Bl79] used the cohomology of the sheaves K_{n} to describe the Abel-Jacobi map CH^{n}(X) → Alb(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 Kcohomology 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 [Bl80].

Homological methods

Most attempts to compute the groups K_{n}(R) in the 1970’s used homological methods. The idea was to use geometric methods to calculate either the homology of the groups SL_{n}(R) or the homology of BQ_{n} for small subcategories Q_{n} of QP(R). If R is the ring of integers in a number field, Quillen [Q73ℤ] used these methods to show that the K_{n}(R) are finitely generated, and Borel [Bo72] used them to compute the rank of the K_{n}(R).

Motivated by these calculations, and also by his calculations in [Ba64], Bass posed this question at the Battelle conference: are the groups K_{n}(R) finitely generated 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.

Here are two of the strongest results in this direction. If R is regular, finitely generated as a ring, and dim(R) ≤ 1, Quillen proved that the K_{n}(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 ℚ-algebra, Remi Kuku [Ku86] proved that each K_{n}(R) is finite.

Several people used homological methods to prove stability theorems for higher K-groups, including Charney, Maazen, Vogtman, van der Kallen and Suslin.

It was sometimes possible to use homological methods to calculate K_{3} and even K_{4}. In 1975, Lee and Szczarba [LeeSz] were able to calculate the homology of SL_{3}(ℤ) and prove that K_{3}(ℤ) is isomorphic to the cyclic group ℤ/48. Evens and Friedlander computed K_{3} and K_{4} for rings of order p^{2} in [EF80]. However, the difficulty of using this method quickly became overwhelming, as Staffeldt’s 1977 thesis [St79] shows. After a ten year effort by Vic Snaith and his students culminated in the book [ALS85], the effort to use these techniques was effectively abandoned.

The Lichtenbaum Conjectures

Let F be a totally real number field F, and R its ring of integers. In 1972, S. Lichtenbaum [Li73] conjectured that the K-theory of R is related to the values of the zeta function ζ_{F} (s). More precisely, for each i ≥ 1 he conjectured that the rational number |ζ_{F} (1 − 2i)| equals the ratio of the orders of the finite groups K_{4i−2}(R) and K_{4i−1}(R).

The first test of this conjecture was R = ℤ and i = 1. Since ζ_{ℚ}(−1) = −1/12 and K_{2}(ℤ) ≅ ℤ/2, Lichtenbaum’s conjecture predicted that K_{3}(ℤ) has 24 elements. After Lee and Szczarba [LeeSz] showed that it has 48 elements, Lichtenbaum amended his conjecture to say that equality holds “up to a power of 2.”

With hindsight, we know that the connection between the K-theory of R and the zeta function of F goes through the étale cohomology of R. To explain this connection, note that the Galois group G of F̄/F acts on the roots of unity μ_{N} and its twists μ_{N}^{⊗i}. For each i there is an integer w_{i}(F) such that the groups H_{et}^{0}(F, μ_{N}^{⊗i}) are cyclic of order w_{i}(F) for all large N. We have already seen that the number w_{2} appears in the Birch-Tate conjecture that #K_{2}(R) is w_{2}(F)|ζ_{F} (−1)|. Another connection was found by Bruno Harris and Graeme Segal, who proved in [HS75] that K_{2i−1}(R) contains a cyclic summand whose order is w_{2}(F) up to a factor of 2.

Lichtenbaum had already made several conjectures in number theory, including a conjecture that for all i and all primes ℓ, the ℓ-part of the integer w_{i}(F)ζ_{F} (i − 1) equals the order of H_{et}^{1}(R[1/ℓ], μ_{ℓ^{ν}}^{⊗i})_{tors} for all large ν. In 1990, Wiles [Wi90] proved the Main Conjecture of Iwasawa Theory for odd primes ℓ, which implies this conjecture up to powers of two. The 2-part of this conjecture would follow from the Main 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 ν large, we have

2^{(?)} · \frac{#K_{2i−2}(R)}{#K_{2i−1}(R)} = ∏_{ℓ} \frac{#H_{et}^{2}(R[1/ℓ]; μ_{ℓ^{ν}}^{⊗i})}{#H_{et}^{1}(R[1/ℓ]; μ_{ℓ^{ν}}^{⊗i})}.

We now know [RW97] that “(?)” is the number r_{1} 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 ℓ invertible in R, analogous to the Atiyah-Hirzebruch spectral sequence, but starting from the étale cohomology groups H_{et}^{p}(R, ℤ_{ℓ}(q/2)), whose abutment would coincide with the ℓ-adic completion of K_{n}(R) at least in degrees n > 1 + dim(R). If R is the ring of integers in a number field (and ℓ is odd), such a spectral sequence would degenerate, yielding the cohomological formulas for K_{n}(R) conjectured by Lichtenbaum.

Quillen made this conjecture explicit when F is an algebraically closed field of characteristic p: the groups K_{2n}(F) should be uniquely divisible, and K_{2n−1}(F) should be a divisible group whose torsion subgroup is ℚ/ℤ[1/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 K-theory with finite coefficients. The groups K_{∗}(R; ℤ/ℓ) were first introduced by W. Browder [Br76], who constructed products on them and showed that the Bott element β ∈ K_{2}(R; ℤ/ℓ) plays a central role. For example, if a finite field 𝔽_{q} has q ≡ 1 (mod ℓ) then K_{∗}(𝔽_{q}; ℤ/ℓ) is the graded commutative ring ℤ/ℓ[x, β]/(x^{2} = 0), where x ∈ K_{1} is the class of a unit generating 𝔽_{q}^{×}.

Soulé [Sou79] constructed étale chern classes K_{n}(R; ℤ/ℓ^{ν}) → H_{et}^{2i−n}(R, μ_{ℓ^{ν}}^{⊗i}), and showed that they are onto in many cases. To do this, he established a product formula, and applied it to elements in K_{∗}(R; ℤ/ℓ) of the form β^{i}∪x with x ∈ K_{j}(R). Soulé showed that the orders of these elements are related to the numerator of the Bernoulli numbers B_{k}/4k, as predicted by Lichtenbaum’s conjectures. For example, Soulé produced an element of order 691 in K_{22}(ℤ).

Inspired by Soulé’s calculations, Eric Friedlander invented an approximation to K-theory he dubbed “étale K-theory.” Dwyer and Friedlander [DF82] constructed a spectral sequence of the kind described by Quillen, converging to étale K-theory. Later on, it was proven that étale K-theory of X is obtained from the ring K_{n}(X; ℤ/ℓ^{ν}) by inverting the Bott element, and that K_{n}(X; ℤ/ℓ^{ν}) agrees with étale K-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 d_{i} of the group K_{2i−1}(R) equals the order of the zero of the zeta function ζ_{F} (s) at s = 1 − i for each i. Indeed, the Hurewicz map K_{∗}(R) → H_{∗}(GL(R); ℝ) induces a map r_{i}: K_{2i−1}(R) → ℝ^{d_{i}}, which we now call the Borel regulator. In 1976, Borel showed [Bo77] that the image of r_{i} is a lattice whose covolume is a rational number times

π^{−d_{i}} lim_{s→1−i} ζ_{F} (s)(s + i − 1)^{d_{i}} = π^{−i[F:ℚ]}\sqrt{D}ζ_{F} (i).

Here d_{i} is the order of the pole of the zeta function ζ_{F} (s) at s = 1 − i, and is either r_{2} or r_{1} + r_{2} depending on i (mod 2).

Next, Bloch gave an intense series of lectures on Borel’s regulator at the University of California at Irvine, followed by some conjectures about possible generalizations to regulators for an elliptic curve E over a number field. Writing up these ideas led him to discover a regulator K_{2}(E) → ℝ^{d}, which was included in the 1977 preprint [Bl77]; Bloch gave a talk on this at the 1978 Helsinki ICM.

In September 1979, Manin asked an undergraduate student in Moscow to give a report on Bloch’s Irvine notes [Bl77]. 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 K-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.

Symbol Calculations

From the beginning, there was an interest in constructing and detecting elements of higher K-theory. It was implicit from the start [Ger72, p. 17] that Quillen’s theory should have products K_{i}(A) ⊗ K_{j}(B) → K_{i+j}(A ⊗ B). Such products were independently constructed by Loday [Lo75], using the space BGL(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_{1}, …, x_{n}} in K_{n}(R), yielding the ring homomorphism K_{∗}^{M}(F) → K_{∗}(F) discussed above. Products were used to construct other elements in K_{n}(R), such as the generalized Dennis-Stein symbols ⟨r_{1}, ..., r_{n}⟩, which are defined whenever 1 − ∏ r_{i} 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 formula H^{p}(X, K_{p}) ≅ CH^{p}(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 K-theory.

To show that these new elements were nonzero, people turned to Chern classes. Gersten had constructed Chern classes with values in Kähler differentials Ω_{R}^{∗} as early as 1970 [Glet] [Ger72, p. 238]. These methods were used by Bloch in [Bl78] to construct Chern classes in crystalline cohomology, as part of an effort to relate K-theory to the slope spectral sequence in crystalline cohomology. Product formulas for the étale chern classes [Sou79] and for the classes in H^{∗}(X, K_{∗}) [Su80] showed that Chern classes on K_{n} typically could not detect elements of exponent (n − 1)!.

In 1976, Keith Dennis [D76] discovered trace maps from K_{n}(R) to the Hochschild homology group HH_{n}(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_{n}(R) was the search for a Mayer-Vietoris sequence for higher K-theory associated to an “excision 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 the relative groups K_{1}(R, I) really depend upon R, so that there was no general extension of the Mayer-Vietoris sequence to K_{2}. Swan also gave a description for the kernel of the surjection K_{1}(R, I) → K_{1}(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 K_{1}(R, S, I), i.e., the preceding term in the appropriate long exact sequence.

The description of the relative group K_{2}(R, I) followed a parallel path. In his 1970 thesis [St71], M. Stein gave one candidate for K_{2}(R, I), the definition used in Milnor’s book [Mi71]. Swan pointed out in [Sw71] that one should really define K_{2}(R, I) to be the third group fitting into the natural long exact sequence for K_{∗}(R) → K_{∗}(R/I), and this relative group was a proper quotient of Stein’s candidate. Keune and Loday independently found a description of K_{2}(R, I) in 1977 [Keu78, Lo78]. Then in 1980, a description of the doubly relative group K_{2}(R, R/J, I) was given by Keune [Keu80] and, again independently, by Guin-Waléry and Loday [GWL].

Computing the K-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_{2}(R, I) was mentioned above [MaSt]. Homological methods had limited success in characteristic p, highlighted by [EF80]. If R contains ℚ, then each K_{n}(R, I) is uniquely divisible, and rational homology methods had more success, including Soulé’s calculation [Sou80] when R = ℚ[ε] and climaxing in Goodwillie’s theorem [Gw86] that K_{n}(R, I) is isomorphic to the cyclic homology group HC_{n−1}(R, I). If R has residual characteristic p, Randy McCarthy [Mc97] has recently shown that K_{n}(R, I) is isomorphic to the topological cyclic homology group T HC_{n−1}(R, I).

Pseudo-isotopies and A(X)

Having settled the h-cobordism question, topologists turned to the question of uniqueness of product structures on M ×I, and studied the topological group P(M) of pseudo-isotopies on M, i.e., diffeomorphisms of M × I which are the identity on M × {0}; a pseudo-isotopy which preserves each level M × {t} is an isotopy.

Cerf proved that if dim(M) > 5 and M is simply connected then every pseudoisotopy is homotopic to an isotopy. Hatcher and Wagoner [HW73] computed π_{0}P(M) when π = π_{1}(M) is nontrivial; it is a sum of two terms, one of which is a quotient of K_{2}(ℤπ) they named W h_{2}(π). Hatcher also discovered a connection between π_{1}P(M) and K_{2}. This led him to construct a stable space W h^{PL}(M) as the direct limit of the spaces Ω^{2}P(M × I^{k}) and prove that if dim(M) ≫ n one has π_{n}P(M) ≅ π_{n}W h^{PL}(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 K-groups appear. For example, they showed that π_{0}P(S^{m+2}M) = K_{−m}(ℤπ_{1}M) if m ≥ 4.

Loday [Lo75] introduced the assembly map H_{n}(BG; 𝕂ℤ) → K_{n}(ℤπ), where the domain is the generalized homology of BG with coefficients in the spectrum 𝕂ℤ for K_{∗}(ℤ), and observed that when n = 2 the cokernel is the Hatcher-Wagoner group W h_{2}(π). This led Loday and Waldhausen [Wa78, p. 228] to define the higher Whitehead groups W h_{n}(π), not in terms of W h^{PL}(X) but as the third term in the natural long exact sequence

⋯ → H_{n}(BG; 𝕂ℤ) ⟶ K_{n}(ℤπ) ⟶ W h_{n}(π) ⟶ H_{n−1}(BG; 𝕂ℤ) → ⋯

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

A(X) ≃ Ω^{∞}S^{∞}(X_{+}) × W h^{PL}(X).

In addition, Waldhausen proved in [Wa78] that π_{∗}A(Bπ) ⊗ ℚ ≅ K_{∗}(ℤπ) ⊗ ℚ.

To do this, Waldhausen first defined the algebraic K-theory of a “ring up to homotopy,” R, using a variation BĜL(R)^{+} of Quillen’s +-construction. For example, if Ω′X is Kan’s simplicial loop group for X, one can form the ring up to homotopy Q[Ω′X] = Ω^{∞}S^{∞}(Ω′X_{+}). Waldhausen defined A(X) as the K-theory space of Q[Ω′X] [Wa78, p. 42].

Waldhausen also gave a K-theoretic construction of A(X) in [Wa78, p. 58], a construction now called Waldhausen K-theory which we will describe shortly.

Here is one nice application that came out of Waldhausen’s construction, showing one way that PL manifold theory is simpler than smooth manifold theory. Consider the two topological groups Homeo_{PL}(D^{n}, ∂D^{n}) and Diff(D^{n}, ∂D^{n}) of PLhomeomorphisms, resp. diffeomorphisms, of the disk which fix the boundary. It is a well-known consequence of the “Alexander Trick” that Homeo_{PL}(D^{n}, ∂D^{n}) is contractible. In contrast, Farrell and Hsiang [FH76] used Waldhausen’s results and Borel’s computation of K_{∗}(ℤ) ⊗ ℚ, together with an analysis of the assembly map, to compute the rational homotopy groups of Diff(D^{n}, ∂D^{n}); they are nonzero.

In 1978–1980, Dwyer, Hsiang and Staffeldt [DHS80, HS82] used rational homotopy theory to compute π_{n}A(X) ⊗ ℚ when π_{1}(X) = 0. They stated their answer in terms of classical invariant theory. Later on, it would be recognized as a harbinger of cyclic homology: π_{n}A(X)⊗ℚ ≅ K_{n}(ℤ)⊕HC_{n−1}(R_{.}), where R_{.} is the differential graded algebra of singular chains on ΩX. The integral version of this result is given in [CCGH].

Also in [Wa78], Waldhausen described a stabilization process, using the Ktheory of a simplicial ring, to construct groups K_{n}^{S}(R, M), called the stable Ktheory of R with coefficients in a bimodule M. Waldhausen constructed a natural map from K_{n}^{S}(R, M) to the Hochschild homology group HH_{n−1}(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 π_{1}P(X) is K_{2}^{S}(ℤπ, M), where π = π_{1}(X) and M = π_{2}(X)[π].

Looking ahead, Goodwillie conjectured that there existed a construction for rings like that used to construct Hochschild homology, and that it would be homotopy equivalent to stable K-theory for any ring. Bökstedt gave such a construction in [Bö85], calling it the topological Hochschild Homology T HH(R) of a ring. This played a key role in later computations such as [BHM]. Finally, Dundas and Mc-Carthy showed in [DM94] that T HH(R; M) ≃ K^{S}(R, M) for any simplicial ring R and bimodule M.

Waldhausen K-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 wS_{.}A and studied the space K(A, w) = Ω|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(R_{f}(X), h), where R_{f}(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 BQA ≃ |wS_{.}A|, so that K_{∗}(A, w) = π_{∗}K(A, w) agrees with Quillen’s K_{∗}(A).

A third important application of Waldhausen’s construction is to the category C_{b}(A) of bounded chain complexes in an exact category A, where w is quasiisomorphism; the classical mapping cylinder of chain complexes satisfies Waldhausen’s cylinder axiom. In fact, Waldhausen’s group K_{0}(C_{b}(A), w) recovers the definition of K_{0}(A) given by Grothendieck in [SGA6].

Waldhausen realized early on that if A is the category of projective R-modules then K(C_{b}(A), w) ≃ K(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].

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Department of Mathematics, Rutgers University, U.S.A.

    E-mail address: weibel@math.rutgers.edu