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Nisnevich descent for K-theory of Deligne-Mumford stacks

Published online by Cambridge University Press:  18 October 2011

Amalendu Krishna  and
Paul Arne Østvær
Amalendu Krishna
Affiliation:
School of Mathematics, Tata Institute of Fundamental Research, Homi Bhabha Road, Colaba, Mumbai,India, amal@math.tifr.res.in
Paul Arne Østvær
Affiliation:
Department of Mathematics, University of Oslo, Norway, paularne@math.uio.no
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Abstract

We show localization, excision and descent theorems for K-theory of Deligne-Mumford stacks. Our approach employs the Nisnevich site which is a complete, regular and bounded cd-structure on the category of such stacks and restricts to the usual Nisnevich site on schemes. By combining excision with a refinement of localization sequences due to Krishna and Töen, we show that K-theory of perfect complexes on tame Deligne-Mumford stacks satisfies Nisnevich descent.

Keywords

Deligne-Mumford stacksK-theory of perfect complexesNisnevich descent
Type
Research Article
Information
Journal of K-Theory , Volume 9 , Issue 2 , April 2012 , pp. 291 - 331
Copyright
Copyright © ISOPP 2011

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References

1.Abramovich, D., Vistoli, A., Compactifying the space of stable maps, J. Amer. Math. Soc. 15 (1) (2001), 27–75.CrossRefGoogle Scholar
2.Artin, M., Grothendieck Topologies, Harvard University Seminar, Spring, (1962).Google Scholar
3.Bondal, A., van den Bergh, M., Generators and representability of functors in commutative and noncommutative geometry, Mosc. Math. J. 3 (1) (2003), 1–36.CrossRefGoogle Scholar
4.Brown, K., Gersten, S., Algebraic K-theory as generalized sheaf cohomology, Lecture Notes in Math. 341, Springer-Verlag, New York, 1973, 85–147.Google Scholar
5.Cisinski, D., Tabuada, G., Non-connective K-theory via universal invariants, math.AG/0903.3717, (2009).Google Scholar
6.Cisinski, D., Krishna, A., Østvær, P. A., The cdh-topology and homotopy invariant K-theory of stacks, In preparation.Google Scholar
7.Cortinas, G., Haesemeyer, C., Sclichting, M., Weibel, C.Cyclic homology, cdh-cohomology and negative K-theory, Ann. of Math. 167 (2) (2008), 549–573.CrossRefGoogle Scholar
8.Deligne, P., Mumford, D., The irreducibility of the space of curves of given genus, Inst. Hautes Etudes Sci. Publ. Math. 36, (1969), 75–109.CrossRefGoogle Scholar
9.Hu, P., Kriz, I., Ormsby, K., Equivariant and real motivic stable homotopy theory, K-theory Preprint Archives no. 952, to appear in Adv. in Math. (2010).Google Scholar
10.Jardine, J., Simplicial presheaves, J. Pure Appl. Algebra 47 (1) (1987), 35–87.CrossRefGoogle Scholar
11.Jardine, J., Generalized étale cohomology theories, Progress in Mathematics, 146. Birkhauser Verlag, Basel, 1997.Google Scholar
12.Joshua, R., Higher intersection theory on algebraic stacks. II, K-Theory 27 (3) (2002), 197–244.CrossRefGoogle Scholar
13.Keller, B., On the cyclic homology of ringed spaces and schemes, Doc. Math. 3 (1998), 231–259.CrossRefGoogle Scholar
14.Knutson, D., Algebraic spaces, Lecture Notes in Mathematics 203, Springer-Verlag, Berlin, 1971.Google Scholar
15.Krishna, A., Perfect complexes on Deligne-Mumford stacks and applications, J. K-Theory 4 (3) (2009), 559–603.CrossRefGoogle Scholar
16.Laumon, G., Moret-Baily, L. Champs algebriques, Ergebnisse der Mathematik und ihrer Grenzgebiete 39 (3) Springer-Verlag, Berlin, (2000).Google Scholar
17.Milne, J., Étale cohomology, Princeton University Press, Princeton, (1980).Google Scholar
18.Neeman, A., The connection between the K-theory localization theorem of Thomason, Trobaugh and Yao and the smashing subcategories of Bousfield and Ravenel, Ann. Sci. Ecole Norm. Sup. 25 (4) (1992), 547–566.CrossRefGoogle Scholar
19.Nisnevich, Y. A., The completely decomposed topology on schemes and associated descent spectral sequences in algebraic K-theory, in: Algebraic K-theory: connections with geometry and topology (Lake Louise, AB, 1987), NATO Adv. Sci. Inst. Ser. C Math. Phys. Sci. 279 (1989), 241–342.Google Scholar
20.Olsson, M., Sheaves on Artin stacks, J. Reine Angew. Math. 603 (2007), 55ÂąV112.Google Scholar
21.Rydh, D., The canonical embedding of an unramified morphism in an étale morphism, Math. Z., to appear, (2010).Google Scholar
22.Rydh, D., Existence and properties of geometric quotients, http://www.math.kth.se/~dary/papers.html, preprint (2011).Google Scholar
23.Serpe, C., Resolution of unbounded complexes in Grothendieck categories, J. Pure Appl. Algebra 177 (1) (2003), 103–112.CrossRefGoogle Scholar
24.Serpe, C., Descent properties of equivariant K-theory, math.AG/1002.2565, (2010).Google Scholar
25.Spaltenstein, N., Resolutions of unbounded complexes, Compositio Math. 65 (1988), 121–154.Google Scholar
26.Thomason, R., Algebraic K-theory of group scheme actions., Algebraic topology and algebraic K-theory (Princeton, N.J., 1983), 539-563, Ann. of Math. Stud. 113, Princeton Univ. Press, Princeton, NJ, 1987.Google Scholar
27.Thomason, R., Equivariant resolution, linearization, and Hilbert's fourteenth problem over arbitrary base schemes, Adv. in Math. 65 (1) (1987), 16–34.CrossRefGoogle Scholar
28.Thomason, R., Equivariant algebraic vs. topological K-homology Atiyah-Segal-style, Duke Math. J. 56 (3) (1988), 589–636.CrossRefGoogle Scholar
29.Thomason, R., Trobaugh, T., Higher algebraic K-theory of schemes and of derived categories, The Grothendieck Festschrift, Vol. III, 247-435, Progr. Math. 88, Birkhauser Boston, Boston, MA, 1990.Google Scholar
30.Toen, B., Théorèmes de Riemann-Roch pour les champs de Deligne-Mumford, K-Theory 18 (1999), 33–76.CrossRefGoogle Scholar
31.Töen, B., Derived Azumaya algebras and generators for twisted derived categories, math.AG/1002.2599, (2010).Google Scholar
32.Vistoli, A., Intersection theory on algebraic stacks and on their moduli spaces, Invent. Math. 97 (1989), 613–670.CrossRefGoogle Scholar
33.Voevodsky, V., Homotopy theory of simplicial sheaves in completely decomposed topology, J. Pure Appl. Algebra 214 (2010), 1384–1398.CrossRefGoogle Scholar
34.Voevodsky, V., Unstable motivic categories in Nisnevich and cdh-topology, J. Pure Appl. Algebra 214 (2010), 1399–1406.CrossRefGoogle Scholar
35.Weibel, C. A., Cyclic homology for schemes, Proc. Amer. Math. Soc. 124 (6) (1996), 1655–1662.CrossRefGoogle Scholar