When Ivorra defines the $l$-adic realization of $S$-motives (i.e. of Voevodsky's motives over a scheme $S$) he demands $l$ to be invertible in $S$. Is this condition really necessary? What happens with $\mathbb{Q}_l$-adic cohomology of schemes if $l$ is not invertible in $S$ (but is not equal to the characteristic of $S$)? Will the Tate twist be invertible?
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asked Dec 19, 2010 at 16:09
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$\begingroup$ If $S$ is a reduced $\mathbf{F}_p$-scheme then the etale sheaves $\mu_{p^n}$ on the etale site of $S$ vanish. So "Tate twist" is not a useful operation there. More generally, plenty of ubiquitous finiteness theorems fail for even constant $p$-torsion coefficients in characteristic $p$ (you know the affine line example). One case where things are OK is for constructible coefficients on a proper scheme over a sep. closed field: that's still finite (ultimately by reducing to constant coefficients on smooth proper curves, where can use Artin-Schreier and coherent cohomology; see Milne's book). $\endgroup$– BCnrdDec 19, 2010 at 20:43
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$\begingroup$ Yes, I know that $p$-adic etale sheaves are pathological in characteristic $p$; I would like to understand what happens with them over arithmetic varieties or just over $\mathbb{Z}$ (for example; i.e. if $p$ is not invertible, but is not $0$ or nilpotent either). $\endgroup$– Mikhail BondarkoDec 19, 2010 at 21:44
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1$\begingroup$ Dear Mikhail: Sorry for misunderstanding. What do you have in mind when you ask if the Tate twist will be "invertible"? Please clarify what coefficients you're considering. Proper base change works in general without restriction on torsion orders, so if $X$ is proper over a strictly henselian local noetherian ring with residue char. $p > 0$ then its etale cohom. with $\mathbf{Z}/(p^n)$-coefficients matches that of the special fiber, and is finite. So if $f:X \rightarrow {\rm{Spec}}(\mathbf{Z}[1/N])$ is proper, then $R^i(f_{\ast})(\mathbf{Z}/(p^n))$ is constructible with the "expected" stalks. $\endgroup$– BCnrdDec 19, 2010 at 23:50
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$\begingroup$ Thank you! I am mostly interested in $\mathbb{Q}_l$-coefficients; yet $\mathbb{Z}_l$-ones are also interesting. I wonder whether the properties of etale cohomology with $\mathbb{Q}_l$-coefficients of $\mathbb{Z}[1/l]$-schemes are substantially distinct from those for (flat) $\mathbb{Z}$-ones. $\endgroup$– Mikhail BondarkoDec 20, 2010 at 0:13
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