# Sheaf isomorphism

Let $f:X\rightarrow Y$ be a morphism of schemes and let $\mathcal{F}$ be a sheaf on $Y$. Then there is a natural map $$\Psi:\mathcal{F} \rightarrow f_{\ast}f^{\ast}(\mathcal{F})$$ and localizing $$\Psi_p:\mathcal{F}_p \rightarrow f_{\ast}f^{\ast}(\mathcal{F})_p=f_{p,\ast}f_p^{\ast}(\mathcal{F}_p)$$

I'm interested in learning under what conditions on $X,Y,f,\mathcal{F},p$ is $\Psi_p$ an isomorphism. E.g. is it an isomorphism if $\mathcal{F}$ is quasi-coherent and $f$ is étale?

Thanks in advance for any insight.

• it's always good to test whatever you care about in the affine case first. For example for $R \to C$, $M \otimes_R C$ is twice as big as $M$ as an $R$-vector space. On the other hand, $\Psi$ is an isomorphism when $f$ is a closed immersion. Also, a sheaf map is an iso if and only if it is an iso on every stalk. Sep 4 '14 at 23:16
• (in case you wonder about the negative vote: this is not a bad question, it's probably just not at research level -- math.stackexchange would be a better place) Sep 4 '14 at 23:17