Functoriality of Sheaf Cohomology

Question

Let $f:X\rightarrow Y$ be a continuous map and $\mathscr F$ be a sheaf of abelian groups on $X$ and $\mathscr G$ be a sheaf on $Y$. Then in general, how to define $H^i(Y,\mathscr G)\rightarrow H^i(X,f^{-1}\mathscr G)$ and $H^i(X,\mathscr F)\rightarrow H^i(Y,f_*\mathscr F)$?(Is this direction correct?)

Answer 1

No, the direction in your second statement is not correct and it goes the other way.

First, there is always the natural restriction map $$ \Gamma(Y, \mathscr{G} ) \rightarrow \Gamma( X, f^{-1} \mathscr{G} ) $$ and since the inverse image functor $ f^{-1} $ is exact, the theory of $ \delta $-functors (read Grothendieck) tells you that there is a natural map of cohomologies $$ H^i(Y, \mathscr{G} ) \rightarrow H^i( X, f^{-1} \mathscr{G} ) $$ Now take $ \mathscr{G} = f_* \mathscr{F} $ to get $$ H^i(Y, f_* \mathscr{F} ) \rightarrow H^i( X, f^{-1} f_* \mathscr{F} ) $$and use adjunction $ f^{-1} f_* \mathscr{F} \rightarrow \mathscr{F} $ to compose the last morphism with $$ H^i( X, f^{-1} f_* \mathscr{F} ) \rightarrow H^i ( X, \mathscr{F} ) $$ to obtain the desired morphism.

As for your question, no I do not think that the second morphism can be obtained by an injective resolution of $ \mathcal{F} $ because pushing forward does not necessarily preserve a resolution. Although instead of the above way, I personally prefer to think of the second morphism as showing up from the Leray Spectral Sequence.