K-Theory of Rings

Introduction

An invariant of a ring that is computable, additive under direct sums, and sensitive to the failure of projective modules to be free is supplied by the theory of the finitely generated projective modules over the ring. $K_0(R)$ is the Grothendieck group of their isomorphism classes under direct sum: it records the stable isomorphism classification of projective modules, measures by its torsion the difference between projective and free, and receives the determinant and the rank as homomorphisms to the units and to $\mathbb{Z}$. Adding the automorphism groups of the projective modules and their elementary subgroups gives $K_1(R)$, whose quotient by the image of $K_1$ of the matrix rings is the Whitehead group and whose relation to the special linear group is the Whitehead lemma. The two functors $K_0$ and $K_1$ are the elementary part of the algebraic K-theory of a ring.

This article develops the Grothendieck group $K_0$ of a ring, its presentation by generators and relations, the split exact sequences of $K_0$ of a product and of a quotient, $K_1$ through the Whitehead lemma and the elementary group, the determinant and rank maps, the Mayer–Vietoris and localisation sequences for $K_0$ and $K_1$, the behaviour of both functors under change of ring and under localisation, and the computation for fields, principal ideal domains, local rings and finite-dimensional algebras. It follows Projective and Injective Modules, The Special Linear Group and the Determinant, Localization and Completion of Modules and Extension of Scalars, and it uses the derived-category language of Derived Categories only in the definition of the higher functors, which are not developed here.

The scope of this article is deliberately narrow, and three boundaries must be stated. First, this is the K-theory of a ring, computed from finitely generated projective modules over a discrete ring; the topological K-theory of vector bundles over a space is a different theory with a different input, it is of Part II, and it is not used here — nothing here depends on it, and the two agree only when a ring is the ring of functions on a space, a comparison that is not available in this Part. Second, the K-theory of operator algebras is likewise a Part II theory over topological algebras, treated elsewhere, and its input is the class of projections in a topological algebra, not the class of finitely generated projective modules. Third, the higher algebraic K-theory groups $K_n(R)$ for $n\ge2$ — the Quillen construction by the plus-construction or by the classifying space of the category of projectives, and the corresponding computations — are likewise outside the scope of this article; this article develops $K_0$ and $K_1$ completely and names $K_n$ for $n\ge2$ only in the closing section, with no result that depends on the higher construction.

Throughout, $R$ is a ring with identity $1\neq0$, commutative unless explicitly stated; $\operatorname{Proj}(R)$ is the set of isomorphism classes of finitely generated projective left $R$-modules; $M_n(R)$ is the ring of $n\times n$ matrices; and $GL_n(R)$, $E_n(R)$, $SL_n(R)$ are the general linear, elementary and special linear groups of The General Linear Group, The Special Linear Group and the Determinant and Transvections and Elementary Transformations. No distance, norm, open set or completion occurs; the words limit, stable and local are algebraic.

The Grothendieck Group $K_0$

Definition

Definition. Let $\operatorname{Proj}(R)$ be the set of isomorphism classes of finitely generated projective left $R$-modules. The group $K_0(R)$ is the abelian group presented by generators $[P]$ for $P \in \operatorname{Proj}(R)$ and relations

$$ [P\oplus Q]=[P]+[Q] $$

for all finitely generated projective $P,Q$. Equivalently, $K_0(R)$ is the Grothendieck group of the monoid $\operatorname{Proj}(R)$ under direct sum; the class of a free module of rank one is written $1=[R]$.

Proposition. Every element of $K_0(R)$ is of the form $[P]-[Q]$ with $P,Q$ finitely generated projective, and $[P]=[Q]$ if and only if there is a finitely generated projective $N$ with $P\oplus N\cong Q\oplus N$.

Proof. A relation $[P\oplus Q]-[P]-[Q]=0$ is a difference of two generators, and a formal difference of generators is the sum of such relations; by induction every element is $[P]-[Q]$. For the second statement, the group presented by the generators and the relations is the localisation of the monoid $\operatorname{Proj}(R)$ at the submonoid of the classes of the free modules; two generators become equal exactly when they become equal after adding a free module to each side, that is, after adding a projective of the form $R^n$ to each.

Example. If $R$ is a field, or a division ring, every finitely generated projective module is free of some rank $n$, the monoid is $\mathbb{N}$, and $K_0(R)\cong\mathbb{Z}$, generated by $[R]$.

Example. If $R$ is a principal ideal domain, every finitely generated projective module is free, so again $K_0(R)\cong\mathbb{Z}$. In particular $K_0(\mathbb{Z})\cong\mathbb{Z}$, generated by the class of $\mathbb{Z}$, even though $\mathbb{Z}$ has a rich module theory: $K_0$ sees only the projective, that is the locally free, part.

Example. For a product $R=R_1\times R_2$ of rings, a finitely generated projective $R$-module is a pair $(P_1,P_2)$ of finitely generated projective modules, and $K_0(R)\cong K_0(R_1)\oplus K_0(R_2)$. By induction the same holds for a finite product. In particular, for a commutative ring $R$ that is a product of two nonzero rings, $K_0(R)$ has $1=[R]=[R_1]+[R_2]$ decomposing as a sum of two nonzero idempotent classes.

Example (a ring with $K_0$ not generated by $[R]$). Let $D$ be a division ring and $R=M_2(D)$. The column space $D^2$ is a finitely generated projective left $R$-module and is simple, and the regular module decomposes as $R\cong D^2\oplus D^2$, so $[R]=2[D^2]$ in $K_0(R)$. Every finitely generated projective left $R$-module is a direct sum of copies of $D^2$ — the idempotents of $M_2(D)$ are the zero and the rank-one projections, and $R e_{11}\cong D^2$ — so $K_0(R)$ is generated by the single class $[D^2]$, with no further relation, and $K_0(M_2(D))\cong\mathbb{Z}$ with $[D^2]$ the generator and $[M_2(D)]=2[D^2]$. In this ring the class of the free module of rank one is twice a generator, so it generates only the even classes and $K_0(M_2(D))$ is not generated by $[M_2(D)]$. This is the basic computation behind Morita invariance.

The Rank Map

Definition. A ring $R$ is connected if its only idempotents are $0$ and $1$. For a connected commutative ring $R$ and a finitely generated projective $P$, the rank $\operatorname{rk}(P)$ is the locally constant integer defined by the rank of the free modules $P_{\mathrm{P}}$ over the local rings $R_{\mathrm{P}}$; algebraically, it is the trace of the identity endomorphism of $P$, a unit of $R$ compared to the free module, and it is constant because $R$ is connected.

Proposition. For a connected commutative ring $R$ the rank defines a homomorphism $\operatorname{rk}:K_0(R)\to\mathbb{Z}$, $\operatorname{rk}([P])=\operatorname{rk}P$, and $\operatorname{rk}(1)=1$. For a field $R$ it is an isomorphism.

Proof. Rank is additive on direct sums, $\operatorname{rk}(P\oplus Q)=\operatorname{rk}P+\operatorname{rk}Q$, so it descends to $K_0$; it takes the value $1$ on $R$. For a field every finitely generated projective is determined by its rank, so the map is injective, and it is surjective since $[R]$ has rank $1$.

Corollary. For a connected commutative ring $R$ there is a split exact sequence

$$ 0\to \widetilde{K}_0(R)\to K_0(R)\xrightarrow{\ \operatorname{rk}\ }\mathbb{Z}\to0, $$

where the reduced group $\widetilde{K}_0(R)$ is the kernel of the rank. For a field or a principal ideal domain, $\widetilde{K}_0(R)=0$. In general $\widetilde{K}_0(R)$ is the group of stable isomorphism classes of projectives of rank zero; equivalently, $[P]=[Q]$ in $\widetilde{K}_0$ if and only if $P\oplus R^n\cong Q\oplus R^n$ for some $n$, with both sides of rank zero.

The group $\widetilde{K}_0(R)$ is the algebraic invariant that detects projectives which are not free: it vanishes exactly when every finitely generated projective module of rank zero is stably free, so it is the obstruction to the classification of projective modules by rank alone.

Functoriality of $K_0$

Remark. For a commutative ring $R$ the determinant of an endomorphism of a finitely generated projective module $P$ is defined through a choice of local basis and lies in $R$; the determinant of an automorphism lies in $R^{\times}$. The free module $P\cong R^n$ has no canonical basis, so the determinant is a homomorphism on the group of automorphisms rather than a function on $K_0$; this is why the determinant belongs to the theory of $K_1$ below.

Proposition. $K_0$ is a contravariant functor on rings: a ring homomorphism $\varphi:R\to S$ makes every finitely generated projective $S$-module a finitely generated projective $R$-module by restriction along $\varphi$, and the induced map $\varphi^*:K_0(S)\to K_0(R)$ is a group homomorphism. Dually, extension of scalars gives a map $K_0(R)\to K_0(S)$: the extension of scalars of a finitely generated projective module is finitely generated projective for an arbitrary $\varphi$, without a flatness hypothesis, and the map $\varphi_*:K_0(R)\to K_0(S)$, $[P]\mapsto[S\otimes_RP]$, is a group homomorphism.

Proof. A finitely generated projective module is a direct summand of a finitely generated free module; restriction of scalars along $\varphi$ preserves finite generation and direct summands, so it preserves projectivity. Extension of scalars preserves free modules and direct summands, hence preserves projectivity, and is additive on direct sums in both variables.

Example. For a ring homomorphism $\varphi:R\to S$ the composite $\varphi_*\varphi^*$ need not be the identity; it is the map $[Q]\mapsto[S\otimes_RQ]$ on $K_0(S)$, which is multiplication by the class of $S$ regarded as an $R$-module only when that class is well defined in $K_0(R)$, which it is not in general. This is the first sign that $K_0$ is only the degree-zero part of a richer functor: the higher K-groups, named at the end, supply the missing compatibility.

$K_1$ and the Whitehead Lemma

The Elementary Group

Definition. Let $GL_n(R)$ be the group of invertible $n\times n$ matrices over $R$ and $E_n(R)$ the subgroup generated by the elementary matrices $e_{ij}(\lambda)=\operatorname{id}+\lambda E_{ij}$ with $i\neq j$ and $\lambda \in R$; the group $E(R)=\bigcup_nE_n(R)$ is the elementary group. The inclusions $GL_n(R)\to GL_{n+1}(R)$, $A\mapsto\operatorname{diag}(A,1)$, give the stable general linear group $GL(R)=\bigcup_nGL_n(R)$, and the stable elementary group $E(R)$ is normal in $GL(R)$.

Theorem (Whitehead lemma). The group $E(R)$ is the commutator subgroup of $GL(R)$: $E(R)=[GL(R),GL(R)]$. Consequently $E(R)$ is normal in $GL(R)$ and $K_1(R)$ is abelian.

Proof. The commutator of two elementary matrices is again elementary, from the identity $$ e_{ij}(a)e_{kl}(b)e_{ij}(a)^{-1}e_{kl}(b)^{-1}=e_{il}(ab)\ \text{if } j=k,\ i\neq l;\qquad =e_{kj}(-ba)\ \text{if } l=i,\ k\neq j;\qquad =1\ \text{otherwise}, $$ so $E(R)\subseteq[GL(R),GL(R)]$. For the reverse inclusion, an invertible matrix whose image in $GL(R)$ is in the commutator subgroup is a product of commutators of invertible matrices; each invertible matrix over $R$ is a product of elementary matrices and a matrix which is a permutation of a diagonal matrix dividing the determinant, and the commutator of two such matrices is elementary because the determinants cancel in a commutator and the permutation part contributes an elementary factor. Hence $[GL(R),GL(R)]\subseteq E(R)$.

Definition. The group $K_1(R)$ is

$$ K_1(R)=GL(R)/E(R)=GL(R)^{\mathrm{ab}}. $$

The Determinant Sequence

Definition. For a commutative ring $R$ the determinant $\det:GL(R)\to R^{\times}$ is the stable determinant, well-defined because $\det(\operatorname{diag}(A,1))=\det A$. It kills elementary matrices, since $\det e_{ij}(\lambda)=1$, so it descends to a homomorphism $\det:K_1(R)\to R^{\times}$.

Proposition. For a commutative ring $R$ there are split exact sequences

$$ 1\to SL(R)\to GL(R)\xrightarrow{\ \det\ }R^{\times}\to1, \qquad 1\to E(R)\to SL(R)\to SK_1(R)\to1, $$

where $SL(R)=\bigcup_nSL_n(R)$ and $SK_1(R)=SL(R)/E(R)$ is the special Whitehead group. If $R$ is a field then $E(R)=SL(R)$ and $SK_1(R)=1$.

Proof. The first sequence is exact by the definition of $SL(R)$ and the existence of diagonal matrices of any determinant in $R^{\times}$. The second is exact by definition of $SK_1$. For a field, every matrix in $SL_n(F)$ is a product of elementary matrices by the reduction of a matrix to the identity by row operations of the first kind, so $E_n(F)=SL_n(F)$.

Example. For a Euclidean domain $R$ — in particular for $\mathbb{Z}$ and for $k[x]$ with $k$ a field — the Euclidean algorithm reduces any matrix in $SL_n(R)$ to the identity by elementary row operations, so $E(R)=SL(R)$, $SK_1(R)=1$, and $K_1(R)\cong R^{\times}$: the determinant is an isomorphism. Hence $K_1(\mathbb{Z})\cong\{\pm1\}$ and $K_1(k[x])\cong k^{\times}$ for a field $k$.

Example. For a division ring $D$ every invertible matrix over $D$ is a product of elementary matrices and a monomial matrix, that is a permutation of an invertible diagonal matrix, a theorem of Dieudonné; the monomial matrices in turn have abelianisation $D^{\times}/[D^{\times},D^{\times}]$, since the permutation part contributes only the sign and the elementary relations absorb it. Hence $K_1(D)\cong D^{\times}/[D^{\times},D^{\times}]$, the abelianisation of the multiplicative group, the Dieudonné determinant giving the identification. For a commutative field this is $F^{\times}$, recovering the previous example. For a commutative local ring $R$ the same reduction gives $K_1(R)\cong R^{\times}$: an invertible matrix over a local ring is a product of elementary matrices and of a matrix that is a permutation of a diagonal matrix, and the determinant identifies $GL(R)/E(R)$ with the group of units.

$K_1$ under Products and Quotients

Proposition. For a product $R=R_1\times R_2$ there is a natural isomorphism $K_1(R)\cong K_1(R_1)\oplus K_1(R_2)$, and the determinant maps compose compatibly.

Proof. A matrix over $R_1\times R_2$ is a pair of matrices, and the same holds for invertibility, since an element of a product is invertible exactly when both components are; the elementary matrices are pairs of elementary matrices, so the quotient splits.

Theorem (Mayer–Vietoris for $K_0$ and $K_1$). Let $R\to R/I$ and $R\to S$ be ring homomorphisms with $R/I$ a quotient, and suppose that the resulting square with $T=S\otimes_RR/I$ is cartesian — that is, $R$ is the fibre product of $S$ and $R/I$ over $T$ — and that at least one of the two maps is surjective. Then there is an exact Mayer–Vietoris sequence

$$ K_1(R)\to K_1(S)\oplus K_1(R/I)\to K_1(T)\xrightarrow{\ \partial\ }K_0(R)\to K_0(S)\oplus K_0(R/I)\to K_0(T). $$

If instead $I$ is a nilpotent ideal of $R$, then the reduction $R\to R/I$ induces an isomorphism $K_0(R)\cong K_0(R/I)$ and a surjection $K_1(R)\to K_1(R/I)$ whose kernel is the image of the group of units of $R$ congruent to $1$ modulo $I$ under the natural map.

Proof. For the nilpotent case, an invertible matrix over $R/I$ lifts entrywise to a matrix over $R$, and its invertibility lifts because an inverse can be constructed by the geometric series $\sum_{k\ge0}(1-A)^k$ when $1-A$ is divisible by a nilpotent; this gives the surjectivity on $K_1$, and the kernel is computed by the units congruent to $1$ modulo $I$. A finitely generated projective over $R/I$ lifts to one over $R$ by the same idempotent-lifting, giving the isomorphism on $K_0$. The Mayer–Vietoris sequence is obtained by applying the Grothendieck-group construction to the cartesian square of categories of finitely generated projective modules; the connecting map $\partial$ is built from the idempotent that measures the failure of a projective module over $T$ to glue to one over $R$.

Example. For a commutative local ring $R$ with maximal ideal $\mathrm{M}$ and residue field $k=R/\mathrm{M}$, every finitely generated projective module is free, so $K_0(R)\cong\mathbb{Z}$, and over a local ring every matrix in $SL_n(R)$ with $n\ge2$ is a product of elementary matrices, so $K_1(R)\cong R^{\times}$ via the determinant. The reduction $R\to k$ therefore induces an isomorphism $K_0(R)\cong K_0(k)$ and a surjection $K_1(R)\cong R^{\times}\to k^{\times}\cong K_1(k)$ with kernel $1+\mathrm{M}$. The kernel is not nilpotent in general — for $R=\mathbb{Z}_p$ it is the infinite group $1+p\mathbb{Z}_p$ — so the nilpotent-ideal statement does not apply, and the reduction on $K_1$ is not an isomorphism.

Localisation Sequences

The Localisation Sequence for $K_0$

Theorem (localisation, $K_0$). Let $R$ be a commutative ring and $S\subseteq R$ a multiplicatively closed subset. There is an exact sequence

$$ K_1(R)\to K_1(S^{-1}R)\xrightarrow{\ \partial\ }K_0(R,S)\to K_0(R)\to K_0(S^{-1}R)\to0, $$

where $K_0(R,S)$ is the Grothendieck group of the abelian category of finitely generated $R$-modules that are $S$-torsion, meaning that every element is annihilated by some element of $S$; the maps are the natural ones. The connecting map $\partial$ sends the class of a projective $S^{-1}R$-module to the class of a finite presentation of it by $S$-torsion modules, and the sequence is exact.

Proof (in outline). The category of finitely generated $S^{-1}R$-modules is a quotient of the category of finitely generated $R$-modules at the Serre subcategory of the $S$-torsion modules, and the Grothendieck-group construction is compatible with this quotient: the sequence records the localisation of the abelian category and the right exactness of the quotient functor. The connecting map $\partial$ sends the class of a projective $S^{-1}R$-module to the class of the finite-length lattice that presents it.

Example. For $R=\mathbb{Z}$ and $S=\mathbb{Z}\setminus\{0\}$ one has $S^{-1}R=\mathbb{Q}$, and $K_0(\mathbb{Z})\cong K_0(\mathbb{Q})\cong\mathbb{Z}$, the map being the identity on the class of the free module of rank one. The group $K_0(\mathbb{Z},S)$ is the Grothendieck group of the finite abelian groups, which is free abelian on the classes $[\mathbb{Z}/p]$ for $p$ prime: the simple finite abelian group $\mathbb{Z}/p$ has $\operatorname{End}=\mathbb{F}_p$, every finite abelian group has a composition series with these factors, and the exact sequences $0\to\mathbb{Z}/p\to\mathbb{Z}/p^k\to\mathbb{Z}/p^{k-1}\to0$ give $[\mathbb{Z}/p^k]=k[\mathbb{Z}/p]$ in the group. The image of $K_1(\mathbb{Q})=\mathbb{Q}^{\times}$ in $K_0(\mathbb{Z},S)$ is the same free abelian subgroup generated by the classes $[\mathbb{Z}/p]$, one for each prime, since $p \in \mathbb{Q}^{\times}$ maps to the class of the cokernel of multiplication by $p$ on $\mathbb{Z}$; the elements $\pm1$, lying in the image of $K_1(\mathbb{Z})=\{\pm1\}$, map to zero. The localisation sequence therefore reads

$$ \{\pm1\}\to\mathbb{Q}^{\times}\to K_0(\mathbb{Z},S)\to\mathbb{Z}\xrightarrow{\ \cong\ }\mathbb{Z}\to0, $$

so the kernel of $K_0(\mathbb{Z},S)\to K_0(\mathbb{Z})$ is the free abelian group on the classes of the prime fields $\mathbb{F}_p$.

The Localisation Sequence for $K_1$

Theorem (localisation, $K_1$, the affine line). Let $R$ be a commutative ring and let $R[t]$ be the polynomial ring. The evaluation $t\mapsto0$ gives a split surjection $K_1(R[t])\to K_1(R)$ whose kernel is the group of nilpotent-class elements; in particular, if $R$ is a regular Noetherian ring then the evaluation at $0$ is an isomorphism $K_1(R[t])\cong K_1(R)$ — this is the homotopy invariance of $K_1$ in this range.

Proof (in outline). The surjectivity and the splitting come from the inclusion $R\to R[t]$; the kernel is computed from the localisation sequence applied to $S=\{1,t,t^2,\dots\}$, which identifies it with the classes coming from the localisation $R[t]\to R[t,t^{-1}]$ and then to $R[t]_{(t)}$, and these classes are the nilpotent ones. In the regular case the higher K-groups that would receive the difference vanish in the relevant degree and the kernel is zero.

Remark. The full statement of homotopy invariance for the higher K-groups is the theorem of Quillen, and it belongs . The version for $K_1$ stated here is proved within the theory of $K_0$ and $K_1$ and needs no higher construction.

Computations

Fields and Division Rings

Theorem. For a division ring $D$ one has $K_0(D)\cong\mathbb{Z}$ and $K_1(D)\cong D^{\times}/[D^{\times},D^{\times}]$. For a field $F$ this gives $K_0(F)\cong\mathbb{Z}$ and $K_1(F)\cong F^{\times}$. In particular $K_1(\mathbb{R})\cong\mathbb{R}^{\times}\cong\{\pm1\}\times\mathbb{R}_{>0}$ and $K_1(\mathbb{F}_p)\cong\mathbb{F}_p^{\times}$, cyclic of order $p-1$.

Proof. Every finitely generated module over a division ring is free, giving $K_0$. The group $GL_n(D)$ is generated by $E_n(D)$ together with the invertible diagonal matrices, and $E(D)=[GL(D),GL(D)]$ by the Whitehead lemma, so $K_1(D)=GL(D)^{\mathrm{ab}}$ is the quotient of the diagonal matrices by elementary relations; the diagonal matrices give $D^{\times}$ and the elementary relations generate the commutator subgroup of $D^{\times}$, so the quotient is $D^{\times}/[D^{\times},D^{\times}]$.

Polynomial Rings and Principal Ideal Domains

Theorem. For a principal ideal domain $R$, $K_0(R)\cong\mathbb{Z}$ and $K_1(R)\cong R^{\times}$. For a field $F$, $K_0(F[t])\cong\mathbb{Z}$ and $K_1(F[t])\cong F^{\times}$. For the integers, $K_0(\mathbb{Z})\cong\mathbb{Z}$ and $K_1(\mathbb{Z})\cong\{\pm1\}$.

Proof. Over a principal ideal domain every finitely generated projective module is free, giving $K_0$; the Euclidean algorithm, available in a principal ideal domain, reduces every matrix in $SL_n(R)$ to the identity by elementary row operations, so $E(R)=SL(R)$ and the determinant gives $K_1(R)\cong R^{\times}$. The polynomial ring over a field is Euclidean with the degree function, whence $K_1(F[t])\cong F^{\times}$.

Finite-Dimensional Algebras

Theorem. Let $A$ be a finite-dimensional algebra over a field $F$, with Jacobson radical $\operatorname{rad}A$ and semisimple quotient $A/\operatorname{rad}A\cong\prod_{i=1}^rM_{n_i}(D_i)$ for division rings $D_i$ finite-dimensional over $F$. Then

$$ K_0(A)\cong\mathbb{Z}^r, \qquad K_1(A)\cong\prod_{i=1}^rD_i^{\times}/[D_i^{\times},D_i^{\times}]. $$

Proof. The radical is nilpotent, so the reduction $A\to A/\operatorname{rad}A$ induces isomorphisms on $K_0$ and $K_1$ by the nilpotent-ideal statement above. For a product of matrix rings over division rings, $K_0$ and $K_1$ are the direct sums of those of the factors by the product formulae, and a matrix ring over a division ring has the $K$-groups of the division ring by Morita invariance, with $K_0(M_n(D))\cong\mathbb{Z}$ and $K_1(M_n(D))\cong K_1(D)$.

Example. For the group algebra $F[G]$ of a finite group $G$ over a field $F$ of characteristic zero, the algebra is semisimple by Maschke's theorem, so $\operatorname{rad}=0$, and $K_0(F[G])$ is free of rank equal to the number of isomorphism classes of simple $F[G]$-modules, which is the number of conjugacy classes of $G$; $K_1(F[G])$ is the product of the $K_1$ of the endomorphism division rings of the simple modules. This makes $K_0$ of a group algebra a representation-theoretic invariant: it is free abelian on the simple modules, so its rank is the number of conjugacy classes of $G$ when the field is a splitting field.

The Whitehead Group

Definition. For a group $G$ the Whitehead group $Wh(G)$ is the quotient of $K_1(\mathbb{Z}[G])$ by the subgroup generated by the images of $GL_1(\mathbb{Z}[G])=\pm G$ under the inclusion; it is the cokernel of the natural map $\pm G\to K_1(\mathbb{Z}[G])$.

Proposition. $Wh(G)$ is a quotient of $K_1(\mathbb{Z}[G])$, is generated by the classes of the units of $\mathbb{Z}[G]$ together with the elementary classes, and vanishes for $G$ trivial, for $G$ free, and for $G$ a free abelian group by the theorem of Bass–Heller–Swan. It is the invariant whose vanishing is the obstruction to the uniqueness of a simple homotopy type; the topological formulation of that statement requires the topology of Part II and is not used here.

Proof. The quotient is by definition a quotient of $K_1(\mathbb{Z}[G])$. For $G$ free abelian, $\mathbb{Z}[G]$ is a Laurent polynomial ring and the Bass–Heller–Swan theorem gives $K_1$ as the direct sum of the units and the elementary part, so the quotient by $\pm G$ is zero; the reference is the standard one.

The Higher Functors, Named Only

Remark. The two functors above are the beginning of a sequence. For $n\ge2$ the group $K_n(R)$ is defined as a homotopy group of a construction applied to the category of finitely generated projective $R$-modules; the two standard constructions are the plus-construction on the classifying space of $GL(R)$, which yields the Quillen K-groups, and the classifying space of the category itself, which yields the same groups and is the source of the agreements with the functors computed here: $K_0(R)$ is the Grothendieck group of the category of finitely generated projectives, and $K_1(R)$ is recovered from the fundamental group of the classifying space abelianised by its elementary subgroup. The definitions, those agreements, the fundamental theorems (resolution, dévissage, localisation, and homotopy invariance in all degrees) and the computations form, which is; nothing in the present article rests on them, and the article is complete at $K_0$ and $K_1$.

Remark. Two further theories carry the name K-theory but take different input. The K-theory of vector bundles over a space and the K-theory of projections in a topological algebra both belong to Part II, where they are treated as well; they presuppose a topology that this Part does not have, and they are not related to the present constructions by any result stated here.

Summary

$K_0(R)$ is the Grothendieck group of the finitely generated projective left $R$-modules under direct sum; every element is $[P]-[Q]$, equality is stable isomorphism after adding a projective to both sides, and for a connected commutative ring the rank gives a split surjection onto $\mathbb{Z}$ whose kernel $\widetilde{K}_0(R)$ vanishes exactly when every finitely generated projective of rank zero is stably free. For a field or a principal ideal domain, $K_0\cong\mathbb{Z}$; for a product, $K_0$ is the direct sum; for $M_n(D)$ over a division ring it is $\mathbb{Z}$ generated by the class of $D^n$.

$K_1(R)$ is $GL(R)$ modulo the elementary group $E(R)$, and the Whitehead lemma identifies $E(R)$ with the commutator subgroup of $GL(R)$, so $K_1$ is abelian. For a commutative ring the determinant surjects $K_1(R)$ onto $R^{\times}$, with kernel the quotient of $SL(R)$ by $E(R)$; for a field, a Euclidean domain and a division ring the determinant is an isomorphism or nearly so, giving $K_1(\mathbb{Z})\cong\{\pm1\}$, $K_1(F)\cong F^{\times}$ and $K_1(D)\cong D^{\times}/[D^{\times},D^{\times}]$. Both functors are additive under products, are unchanged by a nilpotent or radical extension of the ring, and satisfy localisation and Mayer–Vietoris sequences; they make the representation-theoretic data of a finite-dimensional algebra into K-theoretic invariants.

This article has treated the K-theory of a discrete ring, through finitely generated projective modules. The topological K-theory of vector bundles, the K-theory of operator algebras, and the higher algebraic K-groups $K_n$ for $n\ge2$ are separate theories: the first two belong to Part II, where they are treated andand the third is; all three are, and none is used here.

Summary of Notation

Symbol	Meaning
$\operatorname{Proj}(R)$	finitely generated projective left $R$-modules
$[P]$	class of $P$ in $K_0(R)$
$K_0(R)$	Grothendieck group of $\operatorname{Proj}(R)$ under $\oplus$
$\widetilde{K}_0(R)$	reduced group, kernel of the rank
$\operatorname{rk}:K_0(R)\to\mathbb{Z}$	rank homomorphism for a connected commutative ring
$GL_n(R)$, $GL(R)$	general linear group and its stable version
$E_n(R)$, $E(R)$	elementary matrices, elementary group
$SL(R)$	stable special linear group, kernel of $\det$
$K_1(R)=GL(R)/E(R)$	Whitehead group of the ring
$SK_1(R)=SL(R)/E(R)$	special Whitehead group
$Wh(G)$	Whitehead group of a group, quotient of $K_1(\mathbb{Z}[G])$
$e_{ij}(\lambda)$	elementary matrix $\operatorname{id}+\lambda E_{ij}$
$K_n(R)$, $n\ge2$	higher algebraic K-groups, named only
$\mathbb{Z}[G]$	integral group ring of a group $G$
$R^{\times}$	group of units of $R$
$M_n(R)$	ring of $n\times n$ matrices over $R$