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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)].
Reviewer: Michael R. Stein

##### MSC:
 18F25 Algebraic $$K$$-theory and $$L$$-theory (category-theoretic aspects) 14C35 Applications of methods of algebraic $$K$$-theory in algebraic geometry 14C15 (Equivariant) Chow groups and rings; motives 18G30 Simplicial sets; simplicial objects in a category (MSC2010)
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