The Stacks project

Proposition 67.32.2. Let $S$ be a scheme. Let $f : X \to Y$ be a morphism of algebraic spaces over $S$. Let $\mathcal{F}$ be a quasi-coherent sheaf of $\mathcal{O}_ X$-modules. Assume

  1. $Y$ is reduced,

  2. $f$ is quasi-separated,

  3. $f$ is of finite type, and

  4. $\mathcal{F}$ is a finite type $\mathcal{O}_ X$-module.

Then there exists an open dense subspace $W \subset Y$ such that the base change $X_ W \to W$ of $f$ is flat and of finite presentation and such that $\mathcal{F}|_{X_ W}$ is flat over $W$ and of finite presentation over $\mathcal{O}_{X_ W}$.

Proof. This follows immediately from Proposition 67.32.1 and the fact that “of finite presentation” $=$ “locally of finite presentation” $+$ “quasi-compact” $+$ “quasi-separated”. $\square$

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