Let $f: X \rightarrow Y$ be a morphism of varieties, and $\mathscr F$ be a sheaf (mamybe of Abelian groups) over $Y$.
Does $f$ induce a morphism $H^i(Y,\mathscr F) \rightarrow H^i(X,f^{-1}\mathscr F)$? If it does, then how is this morphism defined?
If $f$ is a covering map, or more exactly, a finite Galois covering, is this induced morphism always injective? If not, what additional restrictions are needed to guarantee its injectivity?
Regards.
