The Stacks project

76.22 Openness of the flat locus

This section is analogue of More on Morphisms, Section 37.15. Note that we have defined the notion of flatness for quasi-coherent modules on algebraic spaces in Morphisms of Spaces, Section 67.31.

Theorem 76.22.1. 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 on $X$. Assume $f$ is locally of finite presentation and that $\mathcal{F}$ is an $\mathcal{O}_ X$-module which is locally of finite presentation. Then

\[ \{ x \in |X| : \mathcal{F}\text{ is flat over }Y\text{ at }x\} \]

is open in $|X|$.

Proof. Choose a commutative diagram

\[ \xymatrix{ U \ar[d]_ p \ar[r]_\alpha & V \ar[d]^ q \\ X \ar[r]^ a & Y } \]

with $U$, $V$ schemes and $p$, $q$ surjective and ├ętale as in Spaces, Lemma 65.11.6. By More on Morphisms, Theorem 37.15.1 the set $U' = \{ u \in |U| : p^*\mathcal{F}\text{ is flat over }V\text{ at }u\} $ is open in $U$. By Morphisms of Spaces, Definition 67.31.2 the image of $U'$ in $|X|$ is the set of the theorem. Hence we are done because the map $|U| \to |X|$ is open, see Properties of Spaces, Lemma 66.4.6. $\square$

Lemma 76.22.2. Let $S$ be a scheme. Let

\[ \xymatrix{ X' \ar[r]_{g'} \ar[d]_{f'} & X \ar[d]^ f \\ Y' \ar[r]^ g & Y } \]

be a cartesian diagram of algebraic spaces over $S$. Let $\mathcal{F}$ be a quasi-coherent $\mathcal{O}_ X$-module. Assume $g$ is flat, $f$ is locally of finite presentation, and $\mathcal{F}$ is locally of finite presentation. Then

\[ \{ x' \in |X'| : (g')^*\mathcal{F}\text{ is flat over }Y'\text{ at }x'\} \]

is the inverse image of the open subset of Theorem 76.22.1 under the continuous map $|g'| : |X'| \to |X|$.

Proof. This follows from Morphisms of Spaces, Lemma 67.31.3. $\square$

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