Higher Algebraic K-Theory of Schemes and of Derived Categories

Abstract

In this paper we prove a localization theorem for the A-theory of commutative rings and of schemes, Theorem 7.4, relating the A'-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 [Ql] for A'-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 A'-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 £ under polynomial extensions 9.5, Vorst-van der Kallen theory for NK 9.12, Goodwillie and Ogle-Weibel theorems relating A-theory to cyclic cohomology 9.10, mod £ Mayer-Vietoris for closed covers 9.8, and mod £ comparison between algebraic and topologi-cal A'-theory 11.5 and 11.9. Indeed most known results in A'-theory can be improved by the methods of this paper, by removing now unnecessary regularity, affineness, and other hypotheses. We also develop the higher A'-theory of derived categories, which is an essential tool in the above results. Our techniques here rest on the brilliant work of Waldhausen [W], who has extended and deepened the foundation of A'-theory beyond that laid down by Quillen, allowing it to bear a heavier load. ^partially supported by NSF and the Sloan Foundation.