Algebraic K-theory as generalized sheaf cohomology

Abstract

Let X be a topological space.IN [2] it was shown how one could define "generalized sheaf cohomology" of X; this generalizes ordinary sheaf cohomology, as well as generalized cohomology in the sense of [Ii]. In case X = Spec A, where A is a regular commutative ring, it was announced in [5] that the (Karoubi-Villamayor)K-groups of A could be obtained as generalized sheaf cohomology groups of X. As a consequence, one obtains a "local to global"(or "Atiyah-Hirzebruch") spectral sequence E~q= KP(×,~_q)~ K_(p+q)(A),where ~ -q is the (abelian) sheaf of local K-groups of A.[Note:In the Karoubi-Villamayor notation, one would write ~q and K p+q instead of ~ -q and K_(p+q).]This application of the results of [2] to K-theory requires an improvement of those results. In particular, one needs to eliminate a boundedness assumption on the coefficient sheaves for generalized sheaf cohomology; this can be done provided X is a Noetherian space of finite Krull dimension. It is the purpose of the present paper to present this improvement and to give the application to K-theory.Since the improvement vastly simplifies the generalized sheaf cohomology theory, we give in Sections i and 2 an account of that theory independent of [2].

Singular homology of arithmetic schemes

“…is pseudo-flasque in the sense of [Brown and Gersten 1973], which shows assertion (i). Finally, (ii) follows from the corresponding hypercohomology spectral sequence converging to ‫ވ‬…”

Hypercovers and simplicial presheaves

“…This paper is concerned with the subject of homotopical sheaf theory, as it has developed over time in the articles [I,B,BG,Th,Jo,J1,J2,J3,J4]. Given a fixed Grothendieck site C, one wants to consider contravariant functors F defined on C whose values have a homotopy type associated to them.…”

“…The work of [BG,Jo,J2] brought the use of model categories into this picture. In the most modern of these [J2], Jardine defined a model category structure on presheaves of simplicial sets with the property that the weak equivalences are local in nature.…”

K-Theory of Semi-local Rings with Finite Coefficients and Étale Cohomology

“…Here we denote by H · Zar (X, Y) the global sections of a functorial simplicial cofibrant or flasque resolution of a simplicial sheaf Y, lifting the functor RΓ(X, Y) of [7]. The most convenient is to use the Godement resolution of Y, as in [48, def.…”