The Stacks project

Lemma 62.3.16. Let $f : X \to Y$ be a morphism of schemes which is separated and locally of finite type. Then functor $f_!$ commutes with direct sums.

Proof. Let $\mathcal{F} = \bigoplus \mathcal{F}_ i$. To show that the map $\bigoplus f_!\mathcal{F}_ i \to f_!\mathcal{F}$ is an isomorphism, it suffices to show that these sheaves have the same sections over a quasi-compact object $V$ of $Y_{\acute{e}tale}$. Replacing $Y$ by $V$ it suffices to show $H^0(Y, f_!\mathcal{F}) \subset H^0(X, \mathcal{F})$ is equal to $\bigoplus H^0(Y, f_!\mathcal{F}_ i) \subset \bigoplus H^0(X, \mathcal{F}_ i) \subset H^0(X, \bigoplus \mathcal{F}_ i)$. In this case, by writing $X$ as the union of its quasi-compact opens and using Lemma 62.3.15 we reduce to the case where $X$ is quasi-compact as well. Then $H^0(X, \mathcal{F}) = \bigoplus H^0(X, \mathcal{F}_ i)$ by √Čtale Cohomology, Theorem 59.51.3. Looking at supports of sections the reader easily concludes. $\square$


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