Abstract
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.
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partially supported by NSF and the Sloan Foundation.
to Alexander Grothendieck on his 60th birthday
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References
Adams, J. F., Stable Homotopy and Generalized Homotopy, University of Chicago Press, (1974).
Bass, H., Algebraic K-Theory, Benjamin (1968).
Brinkmann, K.-H., Diplomarbeit, Univ. Bielefeld (1978).
Brown, K., Abstract homotopy theory and generalized sheaf cohomollogy, Trans. Amer. Math. Soc. 186 (1973), 419–458.
Brown, K. and S. Gersten, Algebraic K-theory as generalized sheaf cohomology, Higher K-Theories, Springer Lect. Notes Math 341 (1973), 266–292.
Bousfield, A. K., D. M. Kan, Homotopy Limits, Completions and Localizations, Springer Lect. Notes Math. 304 (1972).
Carter, D. W., Localization in lower algebraic K-theory, Comm. Alg. 8 (1980), 603–622.
Collino, A., Quittais K-theory and algebraic cycles on almost non-singular varieties. Ill. J. Math 25 (1981), 654–666.
Cortazar, J., Rayuela. Editorial Sudamericana Sociedad Anónima (1963), translated as Hopscotch, Random House (1966).
Dwyer, W., and E. Friedlander, Etale K-theory and arithmetic, Trans. Amer. Math. Soc. 292 (1985), 247–280.
Gabber, O., K-theory of henselian local rings and henselian pairs, preprint (1985).
Gabriel, P., Des catégories abéliènnes. Bull. Soc. Math. France 90 (1962), 323–448.
Gersten, S., The localization theorem for projective modules, Comm. Alg. 2 (1974). 307–350.
Gillet, H., Comparison of K-theory spectral sequences, with applications, Algebraic A’-theory: Evanston 1980, Springer Lect. Notes Math. 854 (1981), 141–167.
Gillet, H., Riemann-Roch theorems for higher algebraic K-theory, Adv. Math. 40 (1981). 203–289.
Gillet, H., On the K-theory of surfaces ivith multiple curves, Duke Math. J. 51(1984), 195–233.
Gillet H., The K-theory of twisted complexes. Applications of Algebraic A’-theory to Algebraic Geometry and Number Theory, Contemporary Math. 55 Pt I. Amer. Math Soc. (1986), 159–191.
Gillet, H., R. W. Thomason. The K-theory of strict hensel local rings and a theorem of Sushn, J. Pure Applied Alg. 34 (1984), 241–254.
Giraud, J., Cohomologie Xon Abélienne. Springer-Verlag (1971).
Godement R., Théorie des Faisceaux, Hermann (1964).
Goodwillie, T., Relative algebraic K-theory and cyclic homology, Ann. Math. 124 (1986), 347–402.
Grayson, D., Higher algebraic K-theory II (after Quillen), Algebraic A’-theory: Evanston 1976. Springer Lect. Notes Math. 551 (1976), 217 – 240.
Grayson, D., Exact sequences in algebraic K-theory, Ill. J. Math. 31 (1987), 598–617.
Grayson, D., Localization for flat modules in algebraic K-theory, J. Alg. 61 (1979), 463–496.
Grothendieck, A., Sur quelques points d’algèbre homologique, Tohoku Math. J. 9 (1957), 119–221.
Hartshorne, R., Residues and Duality. Springer Lect. Notes Math. 20 (1966).
Hinich, V., V. Shektman. Geometry of a category of complexes and algebraic K-theory, Duke Math. J. 52 (1985), 399–430.
Jardine, R., Stable homotopy theory of simplicial presheaves, Canad. J. Math. 39 (1987), 733–747.
Jouanolou, J., Une suite exacte de Mayer-Vietoris en K-théorie algébrique, Higher K-theories, Springer Lect. Notes Math. 341 (1973), 293–316.
Karoubi, M., Algèbres de Clifford et K-théorie, Ann. Sci. Ec. Norm. Sup. 1 (1968), 161–270.
Kato, K. S. Saito, Global class field theory of arithmetic schemes, Applications of Algebraic K-theory to Algebraic Geometry and Number Theory, Contemporary Math., 5 Pt. I, Amer. Math. Soc. (1986), 255–331.
Levine, M., Localization on singular varieties, Invent. Math. 91 (1988), 423–464.
Levine, M., Bloch’s formula for singular surfaces, Topology 24 (1985), 165–174.
Loday, J.-L., Cyclic homology: A survey, Banach Center Publications (Warsaw) 18 (1986), 285–307.
May, J. P., Simplicial Objects in Algebraic Topology, van Nostrand (1967).
May, J. P., Pairings of categories and spectra, J. Pure Applied Alg. 19 (1980), 259–282.
Mumford, D., Lectures on Curves on an Algebraic Surface, Princeton Univ. Press (1966).
Nisnevich, Y., Arithmetic and cohomological invariants of semi-simple group schemes and compactifications of locally symmetric spaces, Funct. Anal. Appl., 14 no 1, (1980), 75–76.
Nisnevich, Y., Adeles and Grothendieck topologies, preprint (1982).
Nisnevich, Y., The completely decomposed topology on schemes and associated descent spectral sequences in algebraic K-theory, Algebraic K-Theory: Connections with Geometry and Topology, Kluwer (1989), 241–341.
Ogle, C., On the K-theory and cyclic homology of a square-zero ideal, J. Pure Applied Alg. 46 (1987), 233–248.
Ogle, C., C. Weibel, Relative K-theory and cyclic homology, to appear.
Pedrini, C., C. Weibel, K-theory and Chow groups on singular varieties, Applications of Algebraic K-theory, Contemp. Math. 55 Pt 1 (1986), 339–370.
Quillen, D., Higher algebraic K-theory I, Higher K-theories, Springer Lect. Notes Math. 341 (1973), 85–147.
Quillen, D., Homotopical Algebra, Springer Lect. Notes Math. 43 (1967).
Serre, J. -P., Prolongement des faisceaux analytiques cohérents, Ann. Inst. Fourier 16 (1966), 363–374.
Staffeldt, R., On fundamental theorems of algebraic K-theory, K-Theory 2 (1989), 511–532.
Suslin, A., On the K-theory of algebraically closed fields, Invent. Math. 73 (1983), 241–245.
Suslin, A., On the K-theory of local fields, J. Pure Applied Alg. 34 (1984), 301–318.
Swan, R., K-theory of quadric hypersurfaces, Ann. Math. 122 (1985), 113–153.
Thomason, R. W., Algebraic K- theory and étale cohomology, Ann. Sci. Ec. Norm. Sup. 18 (1985), 437–552.
Thomason, R. W., Homotopy colimits in the category of small categories, Math. Proc. Camb, Philos. Soc. 85 (1979), 91–109.
Thomason, R. W., First quadrant spectral sequences in algebraic K-theory via homotopy colimits, Comm. Alg. 15 (1982), 1589–1668.
Thomason, R. W., Lefschetz-Riemann-Roch theorem and coherent trace formula, Invent. Math. 85 (1986), 515–543.
van der Kallen, W., Descent for K-theory of polynomial rings, Math. Zeit. 191 (1986), 405–415.
Verdier, J.-L., Catégories dérivées, pp. 262–311 of [SGA 4½], see below.
Vorst, T., Localization of the K-theory of polynomial extensions, Math. Ann. 244 (1979), 333–353.
Waldhausen, F., Algebraic K-theory of spaces, Algebraic and Geometric Topology, Springer Lect. Notes Math. 1126 (1985), 318–419.
Weibel, C., Homotopy algebraic K-theory, Contemp. Math. 83 (1989), 461–488.
Weibel, C., Mayer-Vietoris sequences and module structures on NK, Algebraic K-theory: Evanston 1980, Springer Lect. Notes Math. 854 (1981), 466–493.
Weibel, C., Mayer- Vietoris sequences and modp K-theory, Algebraic K-theory: Oberwolfach 1980, Springer Lect. Notes Math. 966 (1982), 390–407.
Weibel, C., Negative K-theory of varieties with isolated singularities, J. Pure Applied Alg. 34 (1984), 331–342.
Weibel, C., A Brown-G ersten spectral sequence for the K-theory of varieties with isolated singularities, Adv. Math 73 (1989), 192–203.
Weibel, C., Module structures on the K-theory of graded rings, J. Alg. 105 (1987), 465–483.
Grothendieck, A., J. Dieudonné, Éléments de géométrie algébrique, Publ. Math. I. H. E. S. Nos. 8, 11, 17, 20, 24, 28, 32 (1961–1967), Grundleheren, 166 (1971), Press. Univ. France, Springer-Verlag.
Grothendieck, A., Revetments étales et groupe fondamental, Springer Lect. Notes Math. 224 (1971).
Grothendieck, A., Cohomologie local des faisceaux cohérents et théorèmes de Lefschetz locaux et globaux, North-Holland (1968).
Artin, M., A. Grothendieck, J.-L. Verdier, Théorie des topos et cohomologie étale des schémas, Springer Lect. Notes Math., 269, 270, 305 (1972–1973).
Deligne, P. et al, Cohomologie étale, Springer Lect. Notes Math. 569 (1977).
Berthelot, P., A. Grothendieck, L. Illusie, Théorie des intersections et théorème de Riemann-Roch, Springer Lect. Notes Math. 225 (1971).
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Thomason, R.W., Trobaugh, T. (2007). Higher Algebraic K-Theory of Schemes and of Derived Categories. In: Cartier, P., Illusie, L., Katz, N.M., Laumon, G., Manin, Y.I., Ribet, K.A. (eds) The Grothendieck Festschrift. Modern Birkhäuser Classics, vol 88. Birkhäuser, Boston, MA. https://doi.org/10.1007/978-0-8176-4576-2_10
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