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The Stacks project

Lemma 59.48.1. Let $\tau \in \{ Zariski, {\acute{e}tale}, smooth, syntomic, fppf\} $. Let $f : X \to Y$ be a monomorphism of schemes. Then the canonical map $f_{big}^{-1}f_{big, *}\mathcal{F} \to \mathcal{F}$ is an isomorphism for any sheaf $\mathcal{F}$ on $(\mathit{Sch}/X)_\tau $.

Proof. In this case the functor $(\mathit{Sch}/X)_\tau \to (\mathit{Sch}/Y)_\tau $ is continuous, cocontinuous and fully faithful. Hence the result follows from Sites, Lemma 7.21.7. $\square$

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