The Stacks project

Lemma 77.7.7. In Situation 77.7.1 suppose given an affine morphism $g : X \to X'$. Set $u' = f_*u : f_*\mathcal{F} \to f_*\mathcal{G}$. Then $F_{u, iso} = F_{u', iso}$, $F_{u, inj} = F_{u', inj}$, $F_{u, surj} = F_{u', surj}$, and $F_{u, zero} = F_{u', zero}$.

Proof. By Cohomology of Spaces, Lemma 69.11.1 we have $g_{T, *}u_ T = u'_ T$. Moreover, $g_{T, *} : \mathit{QCoh}(\mathcal{O}_{X_ T}) \to \mathit{QCoh}(\mathcal{O}_ X)$ is a faithful, exact functor reflecting isomorphisms, injective maps, and surjective maps. $\square$

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