Higher Algebraic K-Theory of Schemes and of Derived Categories

In this paper we prove a localization theorem for the K -theory of commutative rings and of schemes, Theorem 7.4, relating the K -groups of a scheme, of an open subscheme, and of the category of those perfect complexes on the scheme which are acyclic on the open subscheme. The localization theorem of Quillen [Q1] for K ′- or G-theory is the main support of his many results on the G -theory of noetherian schemes. The previous lack of an adequate localization theorem for K -theory has obstructed development of this theory for the fifteen years since 1973. Hence our theorem unleashes a pack of new basic results hitherto known only under very restrictive hypotheses like regularity. These new results include the “Bass fundamental theorem” 6.6, the Zariski (Nisnevich) cohomolog-ical descent spectral sequence that reduces problems to the case of local (hensel local) rings 10.3 and 19.8, the Mayer-Vietoris theorem for open covers 8.1, invariance mod ℓ under polynomial extensions 9.5, Vorst-van der Kallen theory for NK 9.12, Goodwillie and Ogle-Weibel theorems relating K -theory to cyclic cohomology 9.10, mod ℓ Mayer-Vietoris for closed covers 9.8, and mod ℓ comparison between algebraic and topological K -theory 11.5 and 11.9. Indeed most known results in K -theory can be improved by the methods of this paper, by removing now unnecessary regularity, affineness, and other hypotheses. These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.