Higher Algebraic K-Theory of Schemes and of Derived Categories

  • R. W. Thomason
  • Thomas Trobaugh
Part of the Progress in Mathematics book series (MBC, volume 88)


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 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 Mayer-Vietoris for closed covers 9.8, and mod 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.


Exact Sequence Line Bundle Spectral Sequence Abelian Category Exact Functor 


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Copyright information

© Springer Science+Business Media New York 1990

Authors and Affiliations

  • R. W. Thomason
    • 1
  • Thomas Trobaugh
    • 2
  1. 1.Department of MathematicsThe Johns Hopkins UniversityUSA
  2. 2.Université de Paris-SudOrsay CedexFrance

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