Let $X$ be a scheme, $k$ a field and $\sigma \in \mathrm{Aut}(k)$. If $F$ is a sheaf of $k$-vector spaces on $X$ and $u:F\to F$ is a semi-linear endomorphism (meaning $u(\lambda x)=\lambda^\sigma u(x)$ and $u$ is additive), how can I see that the cohomology groups $H^i(X,F)$ are $k$-vector spaces on which $u$ acts as a semi-linear endomorphism ? The case I have in mind is for char($k)=p$, $\sigma : k \to k$ the Frobenius and $u : \mathcal{O}_X \to \mathcal{O}_X$ the $p$-th power map.
If we can compute the cohomology via the Cech complex then this seems clear. Otherwise I've vaguely heard of "hypercoverings" which could settle the question. But I would rather have a solution using the derived functor formalism with which I'm already familar.
EDIT : the flavor of answer I'm looking for would be something like : denote by $\mathrm{SemilinSh}(X)$ the category of sheaves of $k$-vector spaces on $X$ (that is with $k$-linear restrictions), with morphisms all semi-linear morphisms (for all possible $\sigma \in \mathrm{Aut}(k)$, but $\sigma$ is fixed for a given morphism $u :F \to G$, i.e. $u_Y : F(Y) \to G(Y)$ is $\sigma$-semilinear for all $Y$) and $\mathrm{AbSh}(X)$ the category of abelian sheaves on $X$. Then we would like that $\mathrm{SemilinSh}(X)$ has enough injectives, that the forgetful functor $\mathrm{SemilinSh}(X)\to \mathrm{AbSh}(X)$ preserves injectives, and that given $F,G \in \mathrm{SemilinSh}(X)$ with injective resolutions $I^\bullet, J^\bullet$, a $\sigma$-semilinear morphism $F\to G$ has a lift to a $\sigma$-semilinear morphism of complexes $I^\bullet \to J^\bullet$.
Best Answer
The category $\mathrm{Vect}(X)$ of sheaves of $k$-vector spaces identifies with the category of $\underline{k}_X$-modules, hence has enough injectives. Let us show that the forgetful functor $f : \mathrm{Vect}(X) \to \mathrm{Ab}(X)$ to the category of sheaves of abelian groups sends injective objects to $\Gamma$-acyclic objects ($\Gamma$ is the global sections functor).
Let $\mathcal{I}\in\mathrm{Vect}(X)$ be an injective object and let $i : \mathrm{Vect}(X) \to \mathrm{PVect}(X)$ denote the forgetful functor to the category of presheaves of $k$-vector spaces (denote also $i$ the forgetful functor $\mathrm{Ab}(X)\to \mathrm{PAb}(X)$). Then its left adjoint the sheafification functor is exact, so $i$ maps injective objects to injective objects. It follows that for any open cover $\mathcal{U}:U=\bigcup U_i$ in $X$, $i(\mathcal{I})$ is $H^q(\mathcal{U},-)$-acyclic, because $H^q(\mathcal{U},-)$ is the right derived functor of $H^0(\mathcal{U},-)$ in the category $\mathrm{PVect}(X)$ ; but it is also the cohomology of the Cech complex for $\mathcal{U}$ (this is actually the crucial point, and it amounts to showing that the cohomology of the Cech complex vanishes on injective presheaves. I know it is true for presheaves of abelian groups, and I think the proof generalizes, but have not checked it in detail, see Tag01EN). Since the Cech complex of $i(\mathcal{I})$ and $if(\mathcal{I})$ for $\mathcal{U}$ are the same as sets and map of sets, their cohomology is also the same, hence $if(\mathcal{I})$ is $H^q(\mathcal{U},-)$-acyclic for every cover $\mathcal{U}$, so is $\check{H}^q(X,-)$-acyclic. Now the spectral sequence from Cech to derived functor cohomology says that $f(\mathcal{I})$ is $\Gamma$-acyclic.
Let $F \in \mathrm{Vect}(X)$ and fix an injective resolution in $\mathrm{Vect}(X)$ $$0 \to F \to \mathcal{I}^\bullet$$ Then the cohomology abelian groups $H^q(f(\mathcal{I}^\bullet))$ compute $H^q(X,f(F))$ ; but as sets it is the same as the cohomology $k$-vector space $H^q(X,F)$, so there is a natural $k$-vector spaces structure on $H^q(X,f(F))$.
Let now $\sigma \in \mathrm{Aut}(k)$, $F\in\mathrm{Vect}(X)$ and $u : F \to F$ $\sigma$-semilinear. If we denote by $F^\sigma$ the sheaf of $k$-vector spaces $F$ with $k$-action twisted by $\sigma$, then $u$ gives a linear morphism $v:F \to F^\sigma$. The functor $(-)^\sigma : \mathrm{Vect}(X) \to \mathrm{Vect}(X)$ sends injective objects to injective objects, because it is an isomorphism of categories with inverse $(-)^{\sigma^{-1}}$. Therefore if $$0 \to F \to \mathcal{I}^\bullet$$ is an injective resolution, $H^q((\mathcal{I}^\bullet)^\sigma)$ computes $H^q(X,F^\sigma)$. It follows that $H^q(X,F^\sigma)=H^q(X,F)^\sigma$. The linear morphism $v : F \to F^\sigma$ then induces a linear morphism $H^q(X,v) : H^q(X,F)\to H^q(X,F)^\sigma$. The last point is to identify the induced $\sigma$-semilinear morphism $H^q(X,F)\to H^q(X,F)$ with $H^q(X,u)$ ; but as maps of sheaves of abelian groups, $u$ and $v$ are the same and the lift $v':\mathcal{I}^\bullet \to (\mathcal{I}^\bullet)^\sigma$ gives for the underlying belian groups a lift between the $\Gamma$-acyclic resolutions of $F$ of $u=v$, so that as morphisms of abelian groups, $H^q(X,u)=H^q(X,v)$.