Lemma 115.14.1. 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.
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