Nisnevich descent for \(K\)-theory of Deligne-Mumford stacks
type: Item
instance of: scholarly article
zbMATH Open document ID: 1284.19005
author: Amalendu Krishna
author: Paul Arne Østvær
publication date: 2012-05-11T00:00:00Z
full work available at URL: https://arxiv.org/abs/1005.0968
Mathematics Subject Classification ID: 19E08
Mathematics Subject Classification ID: 14C35
Mathematics Subject Classification ID: 14F20
zbMATH DE Number: 6033586
zbMATH Keywords: Deligne-Mumford stacks
zbMATH Keywords: perfect complexes
zbMATH Keywords: Nisnevich descent
MaRDI profile type: Publication
OpenAlex ID: W2325335998
arXiv ID: 1005.0968
cites work: Resolution of unbounded complexes in Grothendieck categories
cites work: The irreducibility of the space of curves of a given genus
cites work: Cyclic homology, cdh-cohomology and negative \(K\)-theory
cites work: Cyclic homology for schemes
cites work: Unstable motivic homotopy categories in Nisnevich and cdh-topologies
cites work: Homotopy theory of simplicial sheaves in completely decomposable topologies
cites work: Intersection theory on algebraic stacks and on their moduli spaces
cites work: Compactifying the space of stable maps
cites work: Riemann-Roch theorems for Deligne-Mumford stacks
cites work: The connection between the $K$-theory localization theorem of Thomason, Trobaugh and Yao and the smashing subcategories of Bousfield and Ravenel
cites work: Q3871006
cites work: Perfect complexes on Deligne-Mumford stacks and applications
cites work: Simplicial presheaves
cites work: Equivariant algebraic vs. topological K-homology Atiyah-Segal-style
cites work: Equivariant resolution, linearization, and Hilbert's fourteenth problem over arbitrary base schemes
DOI: 10.1017/IS011006028JKT161
published in: Journal of \(K\)-Theory
title: Nisnevich descent for \(K\)-theory of Deligne-Mumford stacks
review text: 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. [\textit{K. S. Brown} and \textit{S. M. Gersten}, in: Algebr. K-Theory I, Proc. Conf. Battelle Inst. 1972, Lect. Notes Math. 341, 266--292 (1973; Zbl 0291.18017)] and [\textit{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. [\textit{A. Krishna}, J. K-Theory 4, No. 3, 559--603 (2009; Zbl 1189.19003)] and [\textit{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.
reviewed by: Piotr Krasoń
description: scientific article; zbMATH DE number 6033586