The Stacks project

Lemma 59.48.6. Let $f : X \to Y$ be a closed immersion of schemes. Let $\tau \in \{ syntomic, smooth, {\acute{e}tale}\} $.

  1. The pushforward $f_{big, *} : \mathop{\mathit{Sh}}\nolimits ((\mathit{Sch}/X)_\tau ) \to \mathop{\mathit{Sh}}\nolimits ((\mathit{Sch}/Y)_\tau )$ commutes with coequalizers and pushouts.

  2. The pushforward $f_{big, *} : \textit{Ab}((\mathit{Sch}/X)_\tau ) \to \textit{Ab}((\mathit{Sch}/Y)_\tau )$ is exact.

Proof. This follows from Sites, Lemma 7.42.6, Modules on Sites, Lemma 18.15.3, and Lemma 59.48.5 above. $\square$

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