# Is there a Grothendieck-Riemann-Roch type of theorem generalizing Grothendieck's Lefschetz trace formula

Grothendieck deduced that the L-function of a (constructible) $\ell$-adic sheaf on a variety over $\mathbf{F}_p$ is rational from the generalized trace formula.

My first question is based on the following Remark.

Remark. One can deduce a Lefschetz trace formula with coefficients in $\mathbf{Q}_\ell$ for equivariant varieties from the generalized Lefschetz trace formula with $\mathbf{Q}_{\ell}$ coefficients. (See Proposition 3.2. of the paper Representations of reductive groups over finite fields by Deligne and Lusztig.)

Question 1. Is there a Lefschetz trace formula for equivariant varieties with coefficients in a $\mathbf{Q}_\ell$-sheaf?

My second question is based on the following Theorem and the Remark that follows it. In fact, to prove the generalized trace formula, one proves the more general

Thm. (Local is Global) Let $X = X_0\otimes \overline{\mathbf{F}_p}$, where $X_0$ is a finite type separated $\mathbf{F}_p$-scheme. Let $\Lambda$ be a noetherian torsion ring, killed by a prime $\ell$ which is invertible in $k$. Let $K_0$ be an object of $D^b_{ctf}(X_0,\Lambda)$. Then, for any $n\geq 1$, it holds that $$\sum_{x\in X^{Frob^n}} Tr((Frob^n)^\ast, K_x) = \sum_i (-1)^i Tr((Frob^n)^\ast, R\Gamma_c(X,K)).$$

Here $D^b_{ctf}(X,\Lambda)$, where ctf stands for constructible de Tor dimension finie, is the full subcategory of $D^-(X,\Lambda)$ whose objects are isomorphic to bounded complexes of constructible sheaves of flat $\Lambda$-modules.

Remark. In 1969 Donovan proved a Lefschetz-Grothendieck-Riemann-Roch theorem for equivariant varieties and coherent sheaves with methods as in Borel-Serre's article on GRR.

I now wonder if a Local is Global theorem holds for an equivariant ctf complex over $\mathbf{F}_\ell$. (Here $\ell$ is fixed and I don't know what I mean by an equivariant ctf complex over $\mathbf{F}_\ell$. ) Also, I wonder if a Grothendieck-Riemann-Roch type of generalization holds.

Question 2. Is there a generalization of the Local is Global theorem in the style of Grothendieck-Riemann-Roch?

This could mean the following.

Consider the Grothendieck group of $D^b_{ctf}(X,\Lambda)$. Then for any finite type separated morphism $f:X\longrightarrow Y$, we get a push-forward $Rf_!$ from the Grothendieck group of $X$ to the one of $Y$. Use the $\gamma$-filtration and consider the associated graded object. Then define a Chern character and Todd class in this setting (how?). Does the usual'' diagram commute?

Remark. If such a Grothendieck-Riemann-Roch type theorem can be formulated, then the Local is Global theorem should follow by taking $Y=\textrm{Spec} \ k$.

I'm aware of the fact that this is quite vague. In short, my questions just concern a relativized version of Grothendieck's trace formula to equivariant varieties in the setting of ctf complexes. If we just stick to the Frobenius correspondance then such statements are contained in the work of Grothendieck as written in Deligne's rapport on the subject, I believe.

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+1 for including lots of exposition (I'd also +1 for content, although it's outside my area, and I can't +2). – Theo Johnson-Freyd Nov 21 '10 at 19:21
Thanks! I do think there are some ideas here that are too vague to be taken serious though. Hopefully somebody has some time to say something that will help me. – Ariyan Javanpeykar Nov 22 '10 at 17:16
I would like to point out two references : For question 1. : SGA 5, LNM 589, Exp. IIIA (isn't this the formula you are looking for ?) For question 2. : see Deligne's conjecture, proven by Pink and Fujiwara (see Pink, "On the calculation of the local...", Annals of Math. 135) (sorry if I haven't understood the issues correctly) – Damian Rössler Aug 24 '11 at 14:19