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.

  • 4
    $\begingroup$ 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. $\endgroup$ Sep 4 '14 at 23:16
  • $\begingroup$ (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) $\endgroup$ Sep 4 '14 at 23:17
  • $\begingroup$ I don't understand the notations. You are talking of sheaf but a sheaf in what? sets, abelian groups, $O_Y$-modules? How do you define $f^\ast$ as opposed to $f^{-1}? $\endgroup$
    – Joël
    Sep 4 '14 at 23:40

If $f_*O_X=O_Y$ then this is true for any locally free (or flat) sheaf (and if $f$ is flat then for any quasicoherent sheaf) by projection formula.


The bounded derived category of coherent sheaves is a nice technical tool to understand this question more in depth. Assume that that $A$ and $B$ are bounded complexes of coherent sheaves. Then by using derived functors you get a general version of the projection formula $$Rf_*(B) \otimes^L A \cong Rf_*(B \otimes^L Lf^* A).$$ In order, to deal with your situation you can set $B = \mathcal{O}_X$ and $A = Lf^*\mathcal{F}$. Then you get $$Rf_* (\mathcal{O}_X) \otimes^L Lf^*\mathcal{F} \cong Rf_* Lf^* \mathcal{F}.$$

In order to get the special cases Sasha told you about, you just need to use the fact that $f^*$ is exact if $f$ is flat or that generally $f^*$ is exact on flat sheaves.


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