## 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

### Citations:

Zbl 0291.18017; Zbl 0715.14009; Zbl 1189.19003; Zbl 1275.14017
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### References:

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