The Stacks project

76.10 Flattening in the local case

This section is the analogue of More on Flatness, Section 38.24.

Lemma 76.10.1. Let $S$ be the spectrum of a henselian local ring with closed point $s$. Let $X \to S$ be a morphism of algebraic spaces which is locally of finite type. Let $\mathcal{F}$ be a finite type quasi-coherent $\mathcal{O}_ X$-module. Let $E \subset |X_ s|$ be a subset. There exists a closed subscheme $Z \subset S$ with the following property: for any morphism of pointed schemes $(T, t) \to (S, s)$ the following are equivalent

  1. $\mathcal{F}_ T$ is flat over $T$ at all points of $|X_ t|$ which map to a point of $E \subset |X_ s|$, and

  2. $\mathop{\mathrm{Spec}}(\mathcal{O}_{T, t}) \to S$ factors through $Z$.

Moreover, if $X \to S$ is locally of finite presentation, $\mathcal{F}$ is of finite presentation, and $E \subset |X_ s|$ is closed and quasi-compact, then $Z \to S$ is of finite presentation.

Proof. Choose a scheme $U$ and an ├ętale morphism $\varphi : U \to X$. Let $E' \subset |U_ s|$ be the inverse image of $E$. If $E' \to E$ is surjective, then condition (1) is equivalent to: $(\varphi ^*\mathcal{F})_ T$ is flat over $T$ at all points of $|U_ t|$ which map to a point of $E' \subset |U_ t|$. Choosing $\varphi $ to be surjective, we reduced to the case of schemes which is More on Flatness, Lemma 38.24.3. If $E$ is closed and quasi-compact, then we may choose $U$ to be affine such that $E' \to E$ is surjective. Then $E'$ is closed and quasi-compact and the final statement follows from the final statement of More on Flatness, Lemma 38.24.3. $\square$


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