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Let $X_0$ be a smooth variety (for simplicity I'm willing to assume that X is a curve) over a finite field $k$, $X$ its geometric base change, and $\mathcal{F}$ an $l$-adic etale sheaf on $X$ with $\mathcal{F}(m)$ its Tate-twist.

What is the relation between the weights of Frobenius on $H^r(X,\mathcal{F})$ and $H^r(X,\mathcal{F}(m))$?

I remember seeing somewhere (in Milne?) that $H^r(X,\mathcal{F})\otimes \mathbb{Z}_l(m)\cong H^r(X,\mathcal{F}(m))$ under some restriction to $k$, but I can't find it now. Also, it strikes me as an odd result, considering that the analogue for Serre twists in the classical sheaf cohomology setting doesn't hold.

While searching for an answer in Milne, Etale Cohomology i read about the concept of Purity. Say, I'm willing to assume $\mathcal{F}$ is pure of some weight.

Is there any relation between "Cohomological Purity" in the sense of Milne, namely that something happens in a certain codimension, and the concept of a sheaf being of "pure weight" $s$, i.e. having Frobenius eigenvalues with complex norm $|k|^{s/2}$?

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The isomorphism about Tate twists follows from the projection formula $Rf_*(\mathscr F\otimes^Lf^*\mathscr G)\simeq Rf_*(\mathscr F)\otimes^L\mathscr G.$ Then $\mathscr G$ is flat, there's no need to put $"L"$ there. – shenghao May 6 '13 at 15:43
up vote 6 down vote accepted

You remember correctly, that the effect of Tate twisting the sheaf $\mathcal F$ is just to multiply the Frobenius eigenvalues by a factor of $q$. This result is not surprising because the Tate twist $\mathcal F(m)$ has no relation to the Serre twist on a projective variety, they are just denoted the same way.

The analogous operation to Tate twist in the classical cohomology of a complex variety is defined on the singular cohomology, not in coherent cohomology. There it has the effect of changing the natural $\mathbf Z$-structure on the complex cohomology by a factor $(2\pi i)$, shifting the weight filtration on the mixed Hodge structure by two steps and the Hodge filtration by one step.

Purity in $\ell$-adic cohomology has no direct relation to cohomological purity. But if you interpret a Frobenius eigenvalue of absolute value $q^{k/2}$ as something "k-dimensional", then both are statements that one thing or another is equidimensional, that it has pure dimension.

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