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Let $f \colon Y \to X$ a morphism of schemes. I denote by $f_*$ and $f^*$ the pushforward and pullback of sheaves of modules. Let $U \subseteq X$ be an open (Zariski) set, and denote by $i \colon U \to X$ the inclusion. Also, denote by \begin{equation} f_U \colon f^{-1}(U) \to U \end{equation} the map induced by $f$, and denote by $j \colon f^{-1}(U) \to Y$ the inclusion. I think that it is pretty immediate to show that, if $\mathscr F$ is a sheaf of $\mathscr O_Y$-modules, then we have a "base change" isomorphism \begin{equation} i^*f_* \mathscr F \cong {f_U}_* j^* \mathscr F \end{equation} of sheaves of $\mathscr O_U$-modules. After all, $i$ and $j$ are just inclusions of open sets, so both LHS and RHS above are given by $V \mapsto \mathscr F(f^{-1}(V))$, whenever $V \subseteq U$.

Now, I wonder: what if we work with a different topology, say the étale topology? Assume everything as above but $i \colon U \to X$ an étale map, and interpret $f^{-1}(U)$ as the pullback $Y \times_{X} U$. Does the above "base change" isomorphism still hold? I'd say "yes": if so, is there a reference for this, or even for something more general?

EDIT: I know there is a very powerful (cohomological) flat base change theorem, but it is always stated for quasi-coherent modules, and here I'd really like something basic, namely a direct generalization of what happens in the Zariski topology (which is pretty trivial).

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  • $\begingroup$ You can look for the "flat base change". In case $i : U \longrightarrow X$ is flat, the formula you wrote is even true in the derived category. $\endgroup$ – Libli Apr 30 at 15:27
  • $\begingroup$ I don't really need the result in such generality. Moreover, does the flat base change work for any sheaf of modules? Or just for quasi-coherent modules? And does it need assumptions on $f$? I'd really like to stay basic here... $\endgroup$ – Francesco Genovese Apr 30 at 15:36
  • $\begingroup$ Good point! I didn't notice you were asking for $\mathcal{O}_X$-modules, not just quasi-coherent sheaves of $\mathcal{O}_X$-modules. $\endgroup$ – Libli Apr 30 at 15:38

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