I have a finite étale covering $f \colon Y \rightarrow X$ of degree $d$, where $X$ is smooth projective scheme. I want to know for which sheaf of $\mathcal{O}_X$-modules $\mathcal{M}$ the morphism $H^q(X, \mathcal{M}) \rightarrow H^q(Y, f^*\mathcal{M})$ is injective. I claim that I need $\mathcal{M}$ to be quasi-coherent
It is the composition of: $$H^q(X, \mathcal{M}) \xrightarrow{g} H^q(X, f_*f^*\mathcal{M}) \xrightarrow{h} H^q(Y, f^*\mathcal{M})$$ Since $f$ is finite étale, it is also affine and $\mathcal{M}$ is quasi-coherent $h$ is an isomorphism, which means I just need to show that $g$ is injective and it is enough to show that there exists a morphism $f_*f^* \mathcal{M} \rightarrow \mathcal{M}$ such that the composition: $$\mathcal{M} \rightarrow f_*f^* \mathcal{M} \xrightarrow{\phi} \mathcal{M}$$ is identity on $\mathcal{M}$.
I tried to look locally. Take $U$ an affine open subset of $X$, then $f^{-1}(U)$ is affine in $Y$. $f$ is finite étale, so it is finite locally free, i.e. $\mathcal{O}_Y(f^{-1}(U)) = \mathcal{O}_X(U)^d$
$\mathcal{M}(U) \rightarrow f_*f^*\mathcal{M}(U) = f^*\mathcal{M}(f^{-1}(U)) = f^{-1}\mathcal{M}(f^{-1}(U)) \otimes_{f^{-1}\mathcal{O}_X(f^{-1}(U)} \mathcal{O}_Y(f^{-1}(U))$
By definition this comes from the map between presheaves: $$\mathcal{M}(U) \rightarrow f^{!}\mathcal{M}(f^{-1}(U)) \otimes_{f^{!}\mathcal{O}_X(f^{-1}(U)} \mathcal{O}_Y(f^{-1}(U)) = \mathcal{M}(U) \otimes_{\mathcal{O}_X(U)}\mathcal{O}_X(U)^d = \mathcal{M}(U)^d$$
Using that $f(f^{-1}(U)) = U$ is open. Explicitly the map is: $$ a \mapsto (a, \ldots, a)$$ So the map $trace/d$ is what we need and this map between presheaves induces the desired map between sheaves $f_*f^* \mathcal{M} \rightarrow \mathcal{M}$.
Are the arguments correct?
