Lemma 38.10.9 (05I7)—The Stacks project

Table of contents
Part 2: Schemes
Chapter 38: More on Flatness
Section 38.10: Flat finite type modules, Part I
Lemma 38.10.9 (cite)

Lemma 38.10.9. Let $f : X \to S$ be a morphism which is locally of finite presentation. Let $\mathcal{F}$ be a quasi-coherent $\mathcal{O}_ X$-module of finite type. If $x \in X$ and $\mathcal{F}$ is flat at $x$ over $S$, then $\mathcal{F}_ x$ is an $\mathcal{O}_{X, x}$-module of finite presentation.

Proof. Let $s = f(x)$. By Proposition 38.10.3 there exists an elementary étale neighbourhood $(S', s') \to (S, s)$ such that the pullback of $\mathcal{F}$ to $X \times _ S \mathop{\mathrm{Spec}}(\mathcal{O}_{S', s'})$ is of finite presentation in a neighbourhood of the point $x' \in X_{s'} = X_ s$ corresponding to $x$. The ring map

\[ \mathcal{O}_{X, x} \longrightarrow \mathcal{O}_{X \times _ S \mathop{\mathrm{Spec}}(\mathcal{O}_{S', s'}), x'} = \mathcal{O}_{X \times _ S S', x'} \]

is flat and local as a localization of an étale ring map. Hence $\mathcal{F}_ x$ is of finite presentation over $\mathcal{O}_{X, x}$ by descent, see Algebra, Lemma 10.83.2 (and also that a flat local ring map is faithfully flat, see Algebra, Lemma 10.39.17). $\square$

Comments (2)

Comment #8801 by JJ on February 12, 2024 at 14:10

Shouldn't this work just as well for a morphisim locally of finite type?

If $A \rightarrow B$ is ring map of finite type, $M$ is a finitely generated $B$ -module, and $\mathfrak{p}$ is a prime of $B$ such that $M_\mathfrak{p}$ is flat over $A$ , then take $B'$ a finite polynomial ring surjecting onto $B$ . We still have that $M$ is a finite $B'$ -module, and if $\mathfrak{q}$ is the preimage of $\mathfrak{p}$ in $B'$ , then $M_\mathfrak{q} = M_\mathfrak{p}$ is still flat over $A$ . So proposition 05I5 guarantees that $M_\mathfrak{p}$ is finitely presented over $B'_\mathfrak{q}$ . But then $M_\mathfrak{p}$ is also finitely presented over $B_\mathfrak{p}$ .

Comment #9285 by Stacks project on May 22, 2024 at 10:13

First, in your argument, I would prefer using Lemma 10.5 (the one we're commenting on) instead of Proposition 38.10.3.

The reason I don't want to add this, is that it is sort of the wrong thing to ask for: one should ask for the module $\mathcal{F}_x$ to be "relatively finitely presented" with respect to $\mathcal{O}_{X, x}/\mathcal{O}_{S, f(x)}$ . But we haven't introduced this notion for local homomorphisms of local rings which are essentially of finite type.

This is alluded to in the discussion following Definition 38.4.1. There it is also mentioned that we'll work around this as needed. So if we ever need what your (short) argument shows, then we'll just insert that argument where needed.

Finally, in case $\mathcal{F} = \mathcal{O}_X$ we have Lemma 38.10.10.

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