The Stacks project

113.12 Functor of quotients

Lemma 113.12.1. Let $S = \mathop{\mathrm{Spec}}(R)$ be an affine scheme. Let $X$ be an algebraic space over $S$. Let $q_ i : \mathcal{F} \to \mathcal{Q}_ i$, $i = 1, 2$ be surjective maps of quasi-coherent $\mathcal{O}_ X$-modules. Assume $\mathcal{Q}_1$ flat over $S$. Let $T \to S$ be a quasi-compact morphism of schemes such that there exists a factorization

\[ \xymatrix{ & \mathcal{F}_ T \ar[rd]^{q_{2, T}} \ar[ld]_{q_{1, T}} \\ \mathcal{Q}_{1, T} & & \mathcal{Q}_{2, T} \ar@{..>}[ll] } \]

Then exists a closed subscheme $Z \subset S$ such that (a) $T \to S$ factors through $Z$ and (b) $q_{1, Z}$ factors through $q_{2, Z}$. If $\mathop{\mathrm{Ker}}(q_2)$ is a finite type $\mathcal{O}_ X$-module and $X$ quasi-compact, then we can take $Z \to S$ of finite presentation.

Proof. Apply Flatness on Spaces, Lemma 75.8.2 to the map $\mathop{\mathrm{Ker}}(q_2) \to \mathcal{Q}_1$. $\square$


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