Lemma 38.24.3 (05PI)—The Stacks project

Table of contents
Part 2: Schemes
Chapter 38: More on Flatness
Section 38.24: Flattening in the local case
Lemma 38.24.3 (cite)

Lemma 38.24.3. Let $S$ be the spectrum of a henselian local ring with closed point $s$. Let $X \to S$ be a morphism of schemes 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

$\mathcal{F}_ T$ is flat over $T$ at all points of the fibre $X_ t$ which map to a point of $E \subset X_ s$, and
$\mathop{\mathrm{Spec}}(\mathcal{O}_{T, t}) \to S$ 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. For $x \in X_ s$ denote $Z_ x \subset S$ the closed subscheme we found in Remark 38.24.2. Then it is clear that $Z = \bigcap _{x \in E} Z_ x$ works!

To prove the final statement assume $X$ locally of finite presentation, $\mathcal{F}$ of finite presentation and $Z$ closed and quasi-compact. First, choose finitely many affine opens $W_ j \subset X$ such that $E \subset \bigcup W_ j$. It clearly suffices to prove the result for each morphism $W_ j \to S$ with sheaf $\mathcal{F}|_{X_ j}$ and closed subset $E \cap W_ j$. Hence we may assume $X$ is affine. In this case, More on Algebra, Lemma 15.19.4 shows that the functor defined by (1) is “limit preserving”. Hence we can show that $Z \to S$ is of finite presentation exactly as in the last part of the proof of Theorem 38.24.1. $\square$

The Stacks project

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