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I want to use some results true for non-singular varieties on smooth Deligne-Mumford stacks.

1 -- Concrete example: equivariant pullback for cohomology and K-theory of singular spaces that we could also think of as smooth Deligne-Mumford stacks -- like various moduli spaces. (I'm looking at Chriss and Ginzburg for equivariant K-theory of varieties, and I looked at Edidin and Graham's Equivariant Intersection Theory briefly for Chow groups.) K-theoretic pullback for singular spaces is messy. Can I say these spaces are smooth as DM stacks and use results for nonsingular varieties?

2 -- Meta-question: how does one go about translating results of any kind, but particularly K-theoretic results for nonsingular schemes, to results for smooth Deligne-Mumford stacks? Under what conditions do results carry over? I know this is a very big question; is there a concise answer?

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  • $\begingroup$ 1. "Can I say ...?" I don't know the answer, but of course you have to prove it, before saying or using it. What does Edidin say in their paper? \\ 2. one case: the l-adic (resp. Betti) cohom of proper smooth DM-stacks over char. (resp. char. 0), as for proper smooth varieties, are pure. $\endgroup$
    – shenghao
    Mar 22, 2011 at 19:13
  • $\begingroup$ I don't know so much about stacks, so I don't know how to go about proving versions of, say, localization in K-theory for smooth Deligne-Mumford stacks. There seems to be a lot in the literature about Chow groups... $\endgroup$
    – svk
    Mar 22, 2011 at 22:19
  • $\begingroup$ It would help if you were more specific about what K-theoretic results you wanted to translate. For example, a quick search yielded arXiv preprints by Toën on Riemann-Roch theory, and some other interesting papers that refer to it. Is this a direction of interest? $\endgroup$
    – S. Carnahan
    Mar 24, 2011 at 4:16

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