Algebraic K-theory

2020 Mathematics Subject Classification: Primary: 18F25 Secondary: 18-XX [MSN][ZBL]

A branch of algebra, dealing mainly with the study of the so-called $K$-functors ($K_0, K_1$, etc., cf. $K$-functor); it is a part of general linear algebra. It deals with the structure theory of projective modules and their automorphism groups. To put it more simply, it is a generalization of results obtained on the existence and uniqueness (up to an automorphism) of a basis of a vector space and other group-theoretical facts concerning linear groups over fields. On passing from a field to an arbitrary ring $R$ these theorems usually become invalid, and the Grothendieck group $K_0(R)$ and the Whitehead group $K_1(R)$ are, in a certain sense, a measure of their deviation from being true. Similar generalizations of the structure theorems of linear algebra appear in topology. A vector space can be regarded as a special case of a vector bundle. These objects may be studied with the aid of the homotopy theory of vector bundles and of topological $K$-theory. It is important to note in this connection that a projective module can be regarded as the module of sections of a vector bundle. This explains the choice of the class of projective modules as the object of the theory. Algebraic $K$-theory makes extensive use of the theory of rings, homological algebra, category theory and the theory of linear groups.

Algebraic $K$-theory has two different historical origins, both in the field of geometry. The first is related to certain topological obstructions. The starting point was the introduction of the concept of Whitehead torsion, which is connected with the homotopy equivalence of finite complexes and is an element in the Whitehead group, the latter being some quotient group of the group $K_1(R)$, where $R=\Z\;\Pi$ is the integral group ring of the fundamental group $\Pi$. The next step concerned topological spaces $X$ that are dominated by a finite complex, and their generalized Euler characteristic $\chi(A)$, which is an element of group $K_0(\Z\pi)$. The computation of the Whitehead group and $L$-groups (which is, strictly speaking, an algebraic problem concerning group rings), was in fact one of the first objectives of algebraic $K$-theory. Both $K_2$ and other higher functors have topological applications of the same type (for example, an obstruction to the deformation of a pseudo-isotopy of a closed manifold into an isotopy lies in some quotient group of the group $K_2(\Z\pi)$). Algebraic studies of the Whitehead group began in the 1940s. A related field is the study of the structure of linear groups over arbitrary rings, in particular, the theory of determinants over a skew-field [Ar].

The second origin of algebraic $K$-theory was an algebraic proof of the Riemann–Roch theorem [Ma] and its generalizations by A. Grothendieck in 1957. These considerations involved the introduction of the $K$-functor $K(X)$ as the group of values of a universal additive functor on coherent sheaves on a smooth algebraic variety. Moreover, the previously familiar representation rings, Witt rings (cf. Witt ring) of classes of quadratic forms, etc., turned out to be related constructions. The $K$-functor was then transferred to topology, in which it found numerous applications, and with its help several previously unsolved problems could be dealt with.

It became clear, moreover, that this construction reveals new perspectives in the understanding of old analytical problems (the index problem of elliptic operators), topological problems (extraordinary homology theories), and the theory of group representations. However, the development of algebraic $K$-theory for rings (beginning with the establishment of the correspondence (analogy) between projective, finitely-generated modules and vector bundles) was hindered by the fact that an adequate concept, analogous to that of suspension in topology, was lacking in algebra.

The 1950s and 1960s saw the beginning of the systematic study of projective modules over finite groups, and the development of one of the most important ideas on which algebraic $K$-theory is based — the idea of "stabilization" , the essence of which, roughly speaking, is that general relationships are more clearly manifested on passing to the limit of the dimension of the objects studied (e.g. linear groups or projective modules). Connections were noted between algebraic $K$-theory and the reciprocity laws of the theory of algebraic numbers and algebraic functions; studies were made of problems connected with congruence subgroups (cf. Congruence subgroup) and an algebraic analogue of the Bott periodicity theorem — the theory of polynomial extensions — was obtained.

For a ring $R$ with a unit element, the Grothendieck group $K_0(R)$ is defined as the Abelian group generated by the isomorphism classes of finitely-generated projective $R$-modules, with the defining relation:

$$[P_1]+[P_2] = [P_1\oplus P_2]$$ where $P$ is the class of modules isomorphic to the module $P$. Let $\def\GL{\textrm{GL}} \GL(n,R)$ be the general linear group over $R$, let $$A\mapsto \begin{pmatrix}A&0\\0&1\end{pmatrix}$$ be the imbedding of $\GL(n,R)$ in $\GL(n+1,R)$, let $\GL(R)$ be the direct limit of the groups $\GL(n,R)$, and let $\def\E{\textrm{E}}\E(R)$ be the subgroup in $\GL(R)$ generated by the elementary matrices $\def\l{\lambda} e_{ij}^\l$, i.e. by the matrices that have an element $\l\in R$ at the $(i,j)$-th place and agree with the unit matrix in all other places. $\E(R)$ then coincides with the commutator of $\GL(R)$. The quotient group $\GL(R)/\E(R)$ is denoted by $K_1(R)$, and is known as the Whitehead group. Finally, the Steinberg group $\def\St{\textrm{St}}$ for $n\ge 3$ is defined by the generators $x_{ij}^\l$, $\l\in R$, $1\le i,j\le n$, $i\ne j$, and the relations

$$x_{ij}^\l x_{ij}^\mu = x_{ij}^{\l+\mu},$$

$$[x_{ij}^\l,x_{jl}^\mu] = x_{il}^{\l\mu} \textrm{ if } i\ne l,$$

$$[x_{ij}^\l,x_{kl}^\mu] = 1 \textrm{ if } j\ne k,\; i\ne l,$$ Passing to the direct limit, one obtains the group $\St(R)$ and a natural homomorphism

$$\def\phi{\varphi} \phi:\St(R) \to \E(R),$$ with

$$\phi(x_{ij}^\l) = e_{ij}^\l.$$ The kernel $\ker\phi$ is denoted by $K_2(R)$ (the Milnor group). It coincides with the centre of $St(R)$. Thus, $K_0, K_1$ and $K_2$ are functors from the category of rings into the category of Abelian groups. Each of the functors $K_0$ and $K_1$ can be characterized as a functor from finitely-generated projective modules to Abelian groups that satisfies certain properties and is universal with respect to these properties. Such a "universal" characterization makes it possible to define analogues of the functors $K_0$ and $K_1$ on "sufficiently good" categories. In particular, for the category of Noetherian $R$-modules functors $G_i(R)$ quite close to $K_i(R)$ can be defined.

Examples of the groups $K_i(R)$. Let $R$ be a skew-field and let $R^*$ be its multiplicative group. $K_0(R) = \Z$ is then the group of integers, $K_1(R) = R^*/[R^*,R^*]$; and $K_2(\Z)$ is the cyclic group of order two. If $R$ is a finite field, then $K_2(R)=0$.

An important result in algebraic $K$-theory is the exact Mayer–Vietoris sequence for a Cartesian square. The diagram below represents a Cartesian square of ring homomorphisms in which $f_1$ is an epimorphism;

$$\require{AMSmath} \def\mapright#1{\xrightarrow{#1}} \def\mapdown#1{\Big\downarrow\rlap{\raise2pt{\scriptstyle{#1}}}} \begin{array}{ccc} R & \mapright{h_2} & R_2 \\ \mapdown{h_1} & & \mapdown{f_2} \\ R_1& \mapright{f_1} & R' \end{array}\phantom{h},$$ then there is an exact sequence

$$K_1(R) \to K_1(R_1)\oplus K_1(R_2)\to K_1(R')\to K_0(R)\to K_0(R_1)\oplus K_0(R')\to K_1(R)\to\; \cdots$$

If $f_2$ is also an epimorphism, then the sequence is supplemented by the terms

$$K_2(R)\to K_2(R_1)\oplus K_2(R_2)\to K_2(R')\to K_1(R)\to \; \cdots$$ If $I$ is a two-sided ideal of $R$, then the Mayer–Vietoris sequence makes it possible [Mi] to define the relative functors $K_i(R,I)$, which yield an exact sequence

$$K_2(R,I)\to K_2(R)\to K_2(R/I) \to K_1(R,I)\to K_1(R)\to K_1(R/I)\to K_0(R,I)\to K_)(R)\to K_0(R/I).$$

A fairly complete study has been made of the behaviour of $K$-functors on passing from a ring $R$ to its localization with respect to a central, multiplicatively-closed system. In particular, if certain conditions on $R$ are satisfied, then the following exact sequence has been found for the functor $G_0(R)$:

$$\cup_{s\in S} G_0(R/(s))\to G_0(R)\to G_0(S^{-1}R)\to 0.$$ If $R$ is commutative, $K_0(R)$ becomes a ring with a unit element by introducing the multiplication induced by the tensor product of modules. There exists a split epimorphism of $K_0(R)$ onto the ring $H(R)$ of continuous integer-valued functions (the ring $\Z$ is given the discrete topology) on the spectrum of $R$ (cf. Spectrum of a ring). The kernel of this homomorphism is denoted by $\def\a{\alpha} \tilde K_0(R)$. It is known that $\tilde K_0(R)$ is the nil radical of $K_0(R)$ and, if $R$ is Noetherian and if the dimension of its maximal spectrum is $\a$, then $\tilde K_0(R)^{\a+1} = 0$. If this dimension is at most 1, then $\tilde K_0(R)$ is isomorphic to the Picard group $\textrm{Pic} (K)$.

For arithmetical rings there are finiteness theorems for the functors $K_i(R)$ and $G_i(R)$. In fact, if $A$ is the ring of integers or the ring of polynomials over a finite field, and $R$ is an $R$-order and at the same time an $R$-lattice in a semi-simple finite-dimensional algebra over the field of fractions of a ring $A$, then the groups $K_i(R)$ and $G_i(R)$ are finitely generated ($i=0,1$).

The development of algebraic $K$-theory was stimulated by studies carried out on the problem of congruence subgroups: Do all subgroups of finite index in an arithmetical group contain some congruence subgroup? This question is closely connected with the problem of computing the group $K_1(R,I)$ for ideals $I$ in $R$.

Of the results concerning the stable structure of projective modules one can mention the following theorem: If $R$ is a commutative Noetherian ring whose maximal spectrum has dimension $d$, and $A$ is a module-finite $R$-algebra, then any finitely-generated projective $A$-module $P$ such that

$$\def\fm{\mathfrak{m}} P_\fm \cong A_\fm^{d+1}\oplus Q$$ for all maximal ideals $\fm$ of $R$ is isomorphic to $A\oplus N$ (here $M_\fm$ is the localization of the module $A$ at $\fm$). Another important theorem on the structure of projective modules is the cancellation theorem: Let $R$, $A$ and the module $P$ be as above. Let $Q$ be a finitely-generated projective $A$-module, and let $M$ and $N$ be arbitrary $A$-modules. Then it follows from $Q\oplus P\oplus M\cong Q\oplus N$ that

$$P\oplus M\cong N.$$ The stable rank of a ring $R$ is closely connected with problems of the stable structure of projective modules. Thus, if $R$ is a commutative ring of stable rank smaller than $d$, then

$$K_1(R) \cong \GL(d,R)/\E(d,R).$$ In connection with the theory of induced representations of groups, the functors $K_i$ for group rings have been studied. One of the results of these studies is that if $G$ is a finite group of order $n$ and $C$ is the set of cyclic subgroups of $G$, then the index of the subgroup

$$\sum_{c\in C} \textrm{Im} (K_i(Rc)\to K_i(RG))$$ in $K_i(RG)$ is divisible by $n$ if $i=0,1,2$.

Regarding polynomial ring extensions it is known that if $R$ is a regular ring, then

$$K_0(R[t])\cong K_0(R[t,t^{-1}])\cong K_0(R),$$

$$K_1(R[t])\cong K_1(R).$$ Moreover, the sequence

$$0\to K_1(R)\to K_1(R[t])\oplus K_1(R[t^{-1}])\to K_1(R[t,t^{-1}])\to K_0(R) \to 0$$

is exact for any ring $R$.

One result in the computation of the functor $K_2(R)$ is the theorem of Matsumoto: If $R$ is a field, then $K_2(R)$ is given by the generators $\{a,b\}, a,b\in R^*$ and the relations $\{a,bb'\} = \{a,b\}\{a,b'\},\; \{aa',b\} = \{a,b\}\{a',b\},\; \{a,1-a\} = 1$, where the latter holds for all $a\in R^*, a \ne 1$.

In the 1970s there appeared numerous versions of the definitions of the functors $K_i$ for $i\ge 2$. It has been shown [Ba3] that these theories coincide and yield the classical functors $K_n$ if $n\le 2$. In several cases effective methods of computation for higher $K$-groups were found. The development of unitary $K$-theory ([Ba3], Vol. 3), which studies analogous problems for modules on which quadratic and bilinear forms are defined, also began in that decade.

Algebraic $K$-theoretic ideas and results have become most important in certain parts of functional analysis centering around $C^*$-algebras (cf. $C^*$-algebra). Especially in the form of $KK$-theory (or Kasparov $K$-theory). Cf. e.g. [Cu].

In algebraic geometry there are important connections with the Chow groups (cf. Chow ring).

References

[Ar]	E. Artin, "Geometric algebra", Interscience (1957) MR0082463 Zbl 0077.02101
[At]	M.F. Atiyah, "K-theory. Lectures", Benjamin (1967) MR0224083
[Ba]	H. Bass, "Lectures on topics in algebraic K-theory", Tata Inst. (1966) MR0279159 Zbl 0226.13006
[Ba2]	H. Bass, "Algebraic K-theory", Benjamin (1968) MR0249491 Zbl 0174.30302
[Ma]	Yu.I. Manin, "Lectures on the K-functor in algebraic geometry" Russian Math. Surveys, 24 : 5 (1969) pp. 1–89 Uspekhi Mat. Nauk, 24 : 5 (1969) pp. 3–86 MR0265355 Zbl 0204.21302
[Mi]	J.W. Milnor, "Introduction to algebraic K-theory", Princeton Univ. Press (1971) MR0349811 Zbl 0237.18005
[Sw]	R.G. Swan, "Algebraic K-theory", Springer (1968) MR0245634 Zbl 0193.34601
[SwEv]	R.G. Swan, E.G. Evans, "K-theory of finite groups and orders", Springer (1970) MR0308195 Zbl 0205.32105