The Stacks project

Lemma 77.7.9. In Situation 77.7.8.

  1. The functor $F_{flat}$ satisfies the sheaf property for the fpqc topology.

  2. If $f$ is quasi-compact and locally of finite presentation and $\mathcal{F}$ is of finite presentation, then the functor $F_{flat}$ is limit preserving.

Proof. Part (1) follows from the following statement: If $T' \to T$ is a surjective flat morphism of algebraic spaces over $Y$, then $\mathcal{F}_{T'}$ is flat over $T'$ if and only if $\mathcal{F}_ T$ is flat over $T$, see Morphisms of Spaces, Lemma 67.31.3. Part (2) follows from Limits of Spaces, Lemma 70.6.12 if $f$ is also quasi-separated (i.e., $f$ is of finite presentation). For the general case, first reduce to the case where the base is affine and then cover $X$ by finitely many affines to reduce to the quasi-separated case. Details omitted. $\square$

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