×

Nisnevich descent for \(K\)-theory of Deligne-Mumford stacks. (English) Zbl 1284.19005

The stack \(X\) defined over a noetherian base scheme \(S\) is called a Deligne-Mumford stack if the diagonal map \({\Delta}_{X}: X\rightarrow X {\times}_{S} A\) is representable, quasi-compact and separated and there is an \(S\)-scheme and an étale surjective morphism \(U\rightarrow X.\) Algebraic \(K\)-theory of perfect complexes on schemes is known to satisfy excision, localization and the Mayer-Vietoris (cf. [K. S. Brown and S. M. Gersten, in: Algebr. K-Theory I, Proc. Conf. Battelle Inst. 1972, Lect. Notes Math. 341, 266–292 (1973; Zbl 0291.18017)] and [E. A. Nisnevich, in: Algebraic K-theory: Connections with geometry and topology, Proc. Meet., Lake Louise/Alberta (Can.) 1987, NATO ASI Ser., Ser. C 279, 241–342 (1989; Zbl 0715.14009)]). In the paper authors show the excision and localization properties for \(K\)-theory of Deligne-Mumford stacks. The authors use the Nisnevich site and by combining the excision theorem with a refinement of a localization sequence due to Krishna and Töen (cf. [A. Krishna, J. K-Theory 4, No. 3, 559–603 (2009; Zbl 1189.19003)] and [B. Toën, Invent. Math. 189, No. 3, 581–652 (2012; Zbl 1275.14017)]), they show that \(K\)-theory of perfect complexes on tame Deligne-Mumford stacks with coarse moduli schemes satisfies Nisnevich descent.

MSC:

19E08 \(K\)-theory of schemes
14C35 Applications of methods of algebraic \(K\)-theory in algebraic geometry
14F20 Étale and other Grothendieck topologies and (co)homologies
PDF BibTeX XML Cite
Full Text: DOI arXiv

References:

[1] Rydh, Math. Z. (2010)
[2] DOI: 10.1016/S0022-4049(02)00075-0 · Zbl 1033.18007
[3] DOI: 10.1007/BF02684599 · Zbl 0181.48803
[4] DOI: 10.4007/annals.2008.167.549 · Zbl 1191.19003
[5] DOI: 10.1090/S0002-9939-96-02913-9 · Zbl 0855.19002
[6] Brown, Lecture Notes in Math. 341 pp 85– (1973)
[7] DOI: 10.1016/j.jpaa.2009.11.005 · Zbl 1187.14025
[8] Bondal, Mosc. Math. J. 3 pp 1– (2003)
[9] DOI: 10.1016/j.jpaa.2009.11.004 · Zbl 1194.55020
[10] Artin, Grothendieck Topologies (1962)
[11] DOI: 10.1007/BF01388892 · Zbl 0694.14001
[12] DOI: 10.1090/S0894-0347-01-00380-0 · Zbl 0991.14007
[13] DOI: 10.1023/A:1007791200714 · Zbl 0946.14004
[14] Nisnevich, NATO Adv. Sci. Inst. Ser. C Math. Phys. Sci. 279 pp 241– (1989)
[15] Neeman, Ann. Sci. Ecole Norm. Sup. 25 pp 547– (1992) · Zbl 0868.19001
[16] Milne, Étale cohomology (1980) · Zbl 0433.14012
[17] Laumon, Ergebnisse der Mathematik und ihrer Grenzgebiete 39 (2000)
[18] DOI: 10.1017/is008008021jkt067 · Zbl 1189.19003
[19] Knutson, Lecture Notes in Mathematics 203 (1971)
[20] Keller, Doc. Math. 3 pp 231– (1998)
[21] Joshua, null 27 pp 197– (2002)
[22] Jardine, Progress in Mathematics 146 (1997)
[23] DOI: 10.1016/0022-4049(87)90100-9 · Zbl 0624.18007
[24] Thomason, Progr. Math. 88 (1990)
[25] DOI: 10.1215/S0012-7094-88-05624-4 · Zbl 0655.55002
[26] DOI: 10.1016/0001-8708(87)90016-8 · Zbl 0624.14025
[27] Thomason, Ann. of Math. Stud. 113 (1987)
[28] Spaltenstein, Compositio Math. 65 pp 121– (1988)
[29] Olsson, J. Reine Angew. Math. 603 (2007)
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.