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Let $X$ be a proper variety over a finite field $k$ of characteristic $p>0$, and let $\mathcal F$ be a finite rank $\mathbb F_\ell$ local system on (the etale site of) $X$. Is it true (and, if so, how does one prove) that $$\chi(X,\mathcal F)=\operatorname{rk}\mathcal F\cdot\chi(X,\underline{\mathbb F_\ell}_X)$$ where $\underline{\mathbb F_\ell}_X$ denotes the constant local system on $X$?

Some remarks:

  • If $k=\mathbb C$, then this is true (without assuming properness), but the only proof I know is via analytification and topological arguments. This implies the result for $(X,\mathcal F)$ which lift to characteristic zero, but this leaves out lots of varieties $X$ and local systems $\mathcal F$.

  • Properness is necessary when $\operatorname{char}k>0$. Namely, let $f:\mathbb A^1\to\mathbb A^1$ denote the Artin--Schrier map over $\mathbb F_p$, so $f_\ast\underline{\mathbb F_\ell}_{\mathbb A^1}$ is a local system since $f$ is etale. Now $\operatorname{rk}f_\ast\underline{\mathbb F_\ell}_{\mathbb A^1}=p$, but $\chi(\mathbb A^1,f_\ast\underline{\mathbb F_\ell}_{\mathbb A^1})=\chi(\mathbb A^1,\underline{\mathbb F_\ell}_{\mathbb A^1})$ by the Leray spectral sequence.

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As $X$ is proper, the Swan conductor of $\mathcal{F}$ vanishes. Hence the identity $\chi(X,\mathcal F)=\operatorname{rk}\mathcal F\cdot\chi(X,\underline{\mathbb F_\ell}_X)$ follows from Theorem $4.2.9$ in Kato & Saito's "Ramification theory for varieties over a perfect field ".

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  • $\begingroup$ Thanks! As stated, the theorem you cite only applies when $X$ is smooth. Any idea whether the methods would generalize to the non-smooth case? $\endgroup$ – John Pardon Nov 7 '16 at 18:11
  • $\begingroup$ Kato--Saito cites Illusie in the proof of Theorem 4.2.9. Reading the MR review, it would seem that the answer to my question is actually due to Illusie: ams.org/mathscinet-getitem?mr=629127 $\endgroup$ – John Pardon Nov 7 '16 at 18:15
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    $\begingroup$ @JohnPardon If I recall correctly the argument of that paper is essentially due to Deligne but was written up by Illusie. However even the result of Deligne and Illusie is more general than the situation requires. This follows immediately from the Lefschetz fixed point formula of SGA applied to a finite etale cover on which the bundle trivializes. The Euler characteristic may be computed as a sum of traces of the automorphisms of the bundle. The nontrivial automorphisms have no fixed points, so no contribution, and the contribution of the trivial automorphism depends only on the rank. $\endgroup$ – Will Sawin Nov 8 '16 at 4:37
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    $\begingroup$ @WillSawin Great, just let me ask a stupid question: at which point in that argument is properness used? $\endgroup$ – John Pardon Nov 8 '16 at 20:25
  • $\begingroup$ @JohnPardon The Lefschetz fixed point formula requires the variety to be proper. $\endgroup$ – Will Sawin Nov 9 '16 at 1:27

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