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added nt tag, better wording on artin--schrier map
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John Pardon
John Pardon
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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, since iflet $f:\mathbb A^1\to\mathbb A^1$ denotesdenote the Artin--Schrier map over $\mathbb F_p$, thenso $f_\ast\underline{\mathbb F_\ell}_{\mathbb A^1}$ 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.

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 $\operatorname{char}k>0$, since if $f:\mathbb A^1\to\mathbb A^1$ denotes the Artin--Schrier map over $\mathbb F_p$, then $\operatorname{rk}f_\ast\underline{\mathbb F_\ell}_{\mathbb A^1}=p$ 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.

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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John Pardon
John Pardon
  • 18.7k
  • 3
  • 55
  • 131

Euler characteristic of local system depends only on rank?

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 $\operatorname{char}k>0$, since if $f:\mathbb A^1\to\mathbb A^1$ denotes the Artin--Schrier map over $\mathbb F_p$, then $\operatorname{rk}f_\ast\underline{\mathbb F_\ell}_{\mathbb A^1}=p$ 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.