Lemma 54.69.5 (Extension by zero commutes with base change). Let $f: Y \to X$ be a morphism of schemes. Let $j: V \to X$ be an étale morphism. Consider the fibre product

\[ \xymatrix{ V' = Y \times _ X V \ar[d]_{f'} \ar[r]_-{j'} & Y \ar[d]^ f \\ V \ar[r]^ j & X } \]

Then we have $j'_! f'^{-1} = f^{-1} j_!$ on abelian sheaves and on sheaves of modules.

**Proof.**
This is true because $j'_! f'^{-1}$ is left adjoint to $f'_* (j')^{-1}$ and $f^{-1} j_!$ is left adjoint to $j^{-1}f_*$. Further $f'_* (j')^{-1} = j^{-1}f_*$ because $f_*$ commutes with étale localization (by construction). In fact, the lemma holds very generally in the setting of a morphism of sites, see Modules on Sites, Lemma 18.20.1.
$\square$

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