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This question may be more of a philosophical rather than mathematical nature.

Assume I have a scheme $X$ and an endomorphism $F:X\longrightarrow X$. For instance, $X$ might be of finite type over $\mathbb{F}_{p}$ and $F$ might be the Frobenius, but I do not want to restrict myself to this special case.

If $H^{*}(X,R)$ is a Weil cohomology theory (for instance $R=\mathbb{Q}_{l}$, the $l$-adic cohomology) then Grothendieck, generalizing results of Lefschetz, proved the famous trace formula \begin{equation*} \sum_{x\in X^{F}}\operatorname{tr}(F,X_{x})=\sum_{i\in\mathbb{N}}(-1)^{i+1}\operatorname{tr}(F,H^{i}(X,R))\text{.} \end{equation*}

QUESTION: Do any generalizations of this result exist in case I have two morphisms $F_{1},F_{2}$, or any finite or infinite number of automorphisms $F_{i}:X\longrightarrow X$ and $G=\left<F_{i}\mid i\in I\right>$ is the group of automorphisms of $X$ generated by the $F_{i}$'s. Is there an expression for \begin{equation*} \sum_{x\in X^{G}}\operatorname{tr}(F,X_{x})\text{?} \end{equation*}

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    $\begingroup$ I think the answer is "no", but I don't have a proof it is impossible. I also don't see what this has to do with étale fundamental groups. $\endgroup$
    – Will Sawin
    Feb 27 at 15:29
  • $\begingroup$ The usual Lefschetz formula does not need the assumption that $F$ is an automorphism, if I remember correctly. $\endgroup$
    – Z. M
    Feb 27 at 15:43
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    $\begingroup$ The premise doesn't really make sense to me. First of all, the Frobenius is not unramified. Moreover, there is no trace formula for arbitrary endomorphisms of an arbitrary scheme $X$; at the very least $X$ needs to be of finite type over some base $S$ and the morphism $F \colon X \to X$ needs to be an $S$-morphism. Finally, if the field is not algebraically closed, you should probably say $Rf_*^i(X,\mathscr F)$ (or $H^i(X_{\bar k},\mathscr F)$) instead of $H^i(X,\mathscr F)$, where $f \colon X \to S$ is the structure morphism. $\endgroup$
    – R. van Dobben de Bruyn
    Feb 27 at 15:45
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    $\begingroup$ To add more detail to what R. van Dobben de Bruyn said, the Frobenius in the Galois group gives an unramified automorphism, but the Lefschetz fixed point formula for Frobenius only works because that morphism agrees on closed points and stalks with the geometric Frobenius which is an inseperable morphism. It's very special to that particular pair of morphisms and doesn't generalize in the same way. $\endgroup$
    – Will Sawin
    Feb 27 at 15:55
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    $\begingroup$ On the topological side, when you have two or more commuting endomorphisms of a topological space, the closest analogue of the Lefschetz trace formula is “Smith theory.” There are some analogues in algebraic geometry of Smith theory. $\endgroup$
    – Jason Starr
    Feb 27 at 16:26

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