Nisnevich descent for \(K\)-theory of Deligne-Mumford stacks
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.
scientific article; zbMATH DE number 6033586
scientific article