# Generalized Behrend version for Grothendieck-Lefschetz trace formula

[MOVED HERE FROM MSE.]

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.

ADDED QUESTION: Moreover, why is the sum on the left of the second formula finite when the scheme (or stack) is not of finite type? Behrend speaks about this problem, but I do not find where he solves it, if he does.

• The case of general constructible coefficients is not in Behrend's original 1993 paper as you say. But it's in his 2003 book "Derived $\ell$-adic categories for algebraic stacks". (Also, you don't need $X$ proper in your formula.) – Dan Petersen Jul 7 at 20:06
• For the added question: outside the finite type case there's no reason for anything to converge. Behrend does discuss convergence (in the finite type case) but the problematic part is the right hand side, not the left hand side. Indeed a finite type Artin stack can have nonzero rational cohomology in infinitely many degrees. – Dan Petersen Jul 16 at 8:04
• Here is link to the question on Mathematics: Generalized Behrend version for Grothendieck-Lefschetz trace formula. (It is recommended to link each copy to the other ones when cross-posting.) – Martin Sleziak Jul 16 at 8:08

This is Theorem 4.2 of Shenghao Sun's paper $$L$$-Series of Artin stacks over finite fields, Algebra & Number Theory 6 (2012) pp 47–122, doi:10.2140/ant.2012.6.47, arXiv:1008.3689.

Let $$f:\mathscr X_0\to\mathscr Y_0$$ be a morphism of $$\mathbb F_q$$-algebraic stacks, and let $$K_0\in W^{-,stra}_m(\mathscr X_0,\overline{\mathbb Q}_{\ell})$$ be a convergent complex of sheaves. Then

(i) (Finiteness) $$f_!K_0$$ is a convergent complex of sheaves on $$\mathscr Y_0,$$ and

(ii) (Trace formula) $$c_v(\mathscr X_0,K_0)=c_v(\mathscr Y_0,f_!K_0)$$ for every integer $$v\ge1.$$

Here $$c_v$$ is the sum of the trace of the $$v$$th power of Frobenius. Applying this in the case where $$\mathscr Y_0$$ is a point gets you what you want.

• To amplify on Will's answer, the published version of the paper he mentions is projecteuclid.org/euclid.ant/1513729758 – user141204 Jul 7 at 14:24
• @DavidRoberts: About Lauzières' deleted answer, you wrote (and it is now deleted) that "This answer is a rep-farming exercise by a sockpuppet troll". I'm curious, how did you find that out? I am asking because there is another, legitimate user with the same name on MO, and it is known that this Lauzières person had the habit of creating multiple unregistered users on MO (the post on Dec. 19, 2011). Is the new Lauzières the same as the old Lauzières? – Alex M. Jul 12 at 7:44
• @Alex There are comments on now deleted questions (that 10k+ rep users can still see) pointing to various patterns of behaviour involving this user. There is discussion here: meta.mathoverflow.net/questions/4200/flood-of-similar-new-users – David Roberts Jul 12 at 9:35
• Also, the old Lauziéres account(s) had a history of good interaction, the new account suddenly started asking poorly-motivated questions along the line of the sorts of questions the known sockpuppet accounts used. – David Roberts Jul 12 at 9:40
• @W.Rether The stacks in question have finite type, I believe, making the sum finite. – Will Sawin Jul 15 at 18:46