Higher algebraic \(K\)-theory. I.

*(English)*Zbl 0292.18004
Algebr. \(K\)-Theory I, Proc. Conf. Battelle Inst. 1972, Lect. Notes Math. 341, 85-147 (1973).

In this seminal paper, the author defines higher algebraic \(K\)-groups for certain additive categories and generalizes to the groups most of the classical techniques used to study the Grothendieck group \(K_0\). Given a full additive subcategory \({\mathcal M}\) of an abelian category \({\mathcal A}\) which is closed under extensions in \({\mathcal A}\) define the category \(Q({\mathcal M})\) to have the same objects as \({\mathcal M}\), with a morphism \(M\to M\) being an isomorphism of \(M'\) with a subquotient \(M_i/M_o\) of \(M\), where \(M_o\) and \(M/M_i\) are in \({\mathcal M}\). If the isomorphism classes of objects of \({\mathcal M}\) form a set, the geometric realization of the nerve of \(Q({\mathcal M})\) is called the classifying space, \(BQ({\mathcal M})\); it is determined up to homotopy equivalence. By definition, \(K_i({\mathcal M})=\pi_{i+1}(BQ({\mathcal M}),0)\).

The first section of this paper investigates the homotopy-theoretic properties of these classifying spaces and the maps induced by functors an the underlying categories. Section 2 contains the definition and elementary properties of the \(K\)-groups, including basic exact sequences. Section 3–5 are devoted to proofs of the exactness, resolution, devissage and localization theorems which generalize well-known techniques for studying \(K_0\) and \(K_1\) and provide a first justification for the definition offered in §2.

The second part of this paper, §§6–8, applies the general theory to rings and schemes. For a ring (resp. noetherian ring) \(A\), \(K_i(A)\) (resp. \(K_i'(A)\)) are the \(K\)-groups of the category of finitely generated projective (resp. finitely generated) \(A\)-modules. Among the important results are:

(1) \(K_j(A)\overset\approx\rightarrow K_f'(A)\) is regular noetherian.

(2) \(K_i' (A)\approx K_i'(A[r])\); same for \(K_i\) if \(A\) is regular.

(3) \(K_i'(A[t,t^{-1}])\approx K_i' (A)\oplus K_{i-1}'(A)\); same for \(K_i\) if \(A\) is regular.

For a scheme (resp. noetherian scheme) \(X\), \(K_i(X)\) (resp. \(K_i'(X)\)) are the \(K\)-groups of the category of vector bundles (resp. coherent sheaves) an \(X\). Filtering the category of coherent sheaves by codimension of support yields a spectral sequence

\[ E_i^{pq}= \coprod_{\text{cod}(x)=p} K_{-p-q}(k(x))\Rightarrow K_n'(X). \]

When \(X\) is regular and of finite type over a field, this leads to a proof of Bloch’s formula: \(CH^p(X)=H^p (X,K_p(O_x))\), where \(CH^p(X)\) is the group of codimension \(p\) cycles an \(X\) modulo linear equivalence. – This paper contains proofs of all results announced in [Higher \(K\)-theory for categories with exact sequences, to appear in the Proceedings of the June 1972 Oxford Symposium “New developments in topology”] except for the fact that the groups \(K_i(A)\) introduced here agree with those defined via the \(\text{BGL}(A)^+\) construction of [the author, Actes Congr. internat. Math. 1970, 2, 47–51 (1971; Zbl 0225.18011)].

The first section of this paper investigates the homotopy-theoretic properties of these classifying spaces and the maps induced by functors an the underlying categories. Section 2 contains the definition and elementary properties of the \(K\)-groups, including basic exact sequences. Section 3–5 are devoted to proofs of the exactness, resolution, devissage and localization theorems which generalize well-known techniques for studying \(K_0\) and \(K_1\) and provide a first justification for the definition offered in §2.

The second part of this paper, §§6–8, applies the general theory to rings and schemes. For a ring (resp. noetherian ring) \(A\), \(K_i(A)\) (resp. \(K_i'(A)\)) are the \(K\)-groups of the category of finitely generated projective (resp. finitely generated) \(A\)-modules. Among the important results are:

(1) \(K_j(A)\overset\approx\rightarrow K_f'(A)\) is regular noetherian.

(2) \(K_i' (A)\approx K_i'(A[r])\); same for \(K_i\) if \(A\) is regular.

(3) \(K_i'(A[t,t^{-1}])\approx K_i' (A)\oplus K_{i-1}'(A)\); same for \(K_i\) if \(A\) is regular.

For a scheme (resp. noetherian scheme) \(X\), \(K_i(X)\) (resp. \(K_i'(X)\)) are the \(K\)-groups of the category of vector bundles (resp. coherent sheaves) an \(X\). Filtering the category of coherent sheaves by codimension of support yields a spectral sequence

\[ E_i^{pq}= \coprod_{\text{cod}(x)=p} K_{-p-q}(k(x))\Rightarrow K_n'(X). \]

When \(X\) is regular and of finite type over a field, this leads to a proof of Bloch’s formula: \(CH^p(X)=H^p (X,K_p(O_x))\), where \(CH^p(X)\) is the group of codimension \(p\) cycles an \(X\) modulo linear equivalence. – This paper contains proofs of all results announced in [Higher \(K\)-theory for categories with exact sequences, to appear in the Proceedings of the June 1972 Oxford Symposium “New developments in topology”] except for the fact that the groups \(K_i(A)\) introduced here agree with those defined via the \(\text{BGL}(A)^+\) construction of [the author, Actes Congr. internat. Math. 1970, 2, 47–51 (1971; Zbl 0225.18011)].

Reviewer: Michael R. Stein