# “Base change” along étale map

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 $$$$f_U \colon f^{-1}(U) \to U$$$$ the map induced by $$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$$-modules, then we have a "base change" isomorphism $$$$i^*f_* \mathscr F \cong {f_U}_* j^* \mathscr F$$$$ of sheaves of $$\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).

• 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. – Libli Apr 30 at 15:27
• 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... – Francesco Genovese Apr 30 at 15:36
• Good point! I didn't notice you were asking for $\mathcal{O}_X$-modules, not just quasi-coherent sheaves of $\mathcal{O}_X$-modules. – Libli Apr 30 at 15:38