# Pullback and Pushforward Isomorphism of Sheaves

Suppose we have 2 schemes $X, Y$ and also a map $f\colon X\to Y$. After that we understand that $\operatorname{Hom}_X(f^*\mathcal{G}, \mathcal{F})\simeq \operatorname{Hom}_Y(\mathcal{G}, f_*\mathcal{F})$, where $\mathcal{F}$ is an $\mathcal{O}_X$ - component and also $\mathcal{G}$ an $\mathcal{O}_Y$ - component (and also the Homs remain in the group of $\mathcal{O}_X$ - components etc). This offers an all-natural map $f^* f_* \mathcal{F}\to \mathcal{F}$, simply by establishing $\mathcal{G}=f_* \mathcal{F}$ and also considering where the identification map goes.

Exist any kind of well - well-known problems on the map or sheaves that offer this is an isomorphism? As an example, I was browsing a publication and also saw that the map is surjective if $\mathcal{F}$ is a really enough invertible sheaf (and also possibly some even more theory on the map and also $X$ and also $Y$ were thought too).

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