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The statement of the Grothendieck-Lefschetz fixed point theorem is well-known. For a proper algebraic variety $X$ over $\mathbb F_q$, $$\#X(\mathbb F_q) =\sum_i (−1)^i Tr(Fr_X, H^i_c(X, \mathbb Q_l)).$$ Also known is the version for general constructible l-adic sheaves $\mathcal F$: $$\sum_{x\in X(\mathbb F_q)} Tr(Fr_x,\mathcal F_x)=\sum_i (−1)^i Tr(Fr_X, H^i_c(X, \mathcal F)).$$ Thirdly, K. Behrend proved an analog for the first formula in the context of algebraic stacks (replacing the scheme $X$ by a Noetherian algebraic stack $\mathcal X$).

Now my question is: is there a version of the second formula for an algebraic stack $\mathcal X$ (with nice hypotheses if necessary)?

It would seem natural, since the second formula is a generalization of the first, and the first is true in the context of algebraic stacks by Behrend's work. However, the second formula does not follow directly from the first in the case of schemes (as far as I know: I would be glad if it were true!), so I am not in the position to easily extend the proof of the second formula in the more general context of stacks.

Thank you in advance.

W.Rether
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    Interesting question! By choosing a stratification $U_i$ of $X$ where $\mathcal{F}$ be comes locally constant and using inclusion-exclusion, it suffices to handle case that $\mathcal{F}$ is locally constant. Maybe then one can reduce to the case of finite coefficients and $\mathcal{F}$ that becomes constant after a Galois cover? – Pol van Hoften Jul 07 '19 at 07:42
  • Nice clue. One should beware that constructible sheaves on stacks have a slightly more complicated definition: what would we mean by stratification of a stack? – W.Rether Jul 07 '19 at 10:59
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    I realize that the question should be asked on MO. I have therefore asked it there at https://mathoverflow.net/questions/335631/generalized-behrend-version-for-grothendieck-lefschetz-trace-formula. – W.Rether Jul 07 '19 at 14:08

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