The Stacks project

Lemma 107.5.4. Let $B$ be an algebraic space. Let $\pi : X \to Y$ be an affine quasi-finite morphism of algebraic spaces which are separated and of finite presentation over $B$. Let $\mathcal{F}$ be a quasi-coherent $\mathcal{O}_ X$-module. Then $\pi _*$ induces a morphism $\mathrm{Quot}_{\mathcal{F}/X/B} \to \mathrm{Quot}_{\pi _*\mathcal{F}/Y/B}$.

Proof. Set $\mathcal{G} = \pi _*\mathcal{F}$. Since $\pi $ is affine we see that for any scheme $T$ over $B$ we have $\mathcal{G}_ T = \pi _{T, *}\mathcal{F}_ T$ by Cohomology of Spaces, Lemma 68.11.1. Moreover $\pi _ T$ is affine, hence $\pi _{T, *}$ is exact and transforms quotients into quotients. Observe that a quasi-coherent quotient $\mathcal{F}_ T \to \mathcal{Q}$ defines a point of $\mathrm{Quot}_{X/B}$ if and only if $\mathcal{Q}$ defines an object of $\mathcal{C}\! \mathit{oh}_{X/B}$ over $T$ (similarly for $\mathcal{G}$ and $Y$). Since we've seen in Lemma 107.4.4 that $\pi _*$ induces a morphism $\mathcal{C}\! \mathit{oh}_{X/B} \to \mathcal{C}\! \mathit{oh}_{Y/B}$ we see that if $\mathcal{F}_ T \to \mathcal{Q}$ is in $\mathrm{Quot}_{\mathcal{F}/X/B}(T)$, then $\mathcal{G}_ T \to \pi _{T, *}\mathcal{Q}$ is in $\mathrm{Quot}_{\mathcal{G}/Y/B}(T)$. $\square$

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