TheoremStatement: AI-adaptedProof: AI-adaptedprecheck passjudge pass (gpt-5.6-terra)audited 2026-09-01

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For an $R$ -finite module over a local map, flatness on the closed fibre plus the multiplication-map condition implies flatness

Statement

Let $R \to S$ be a local homomorphism of Noetherian local rings with maximal ideal $m \subset R$ , and let $M$ be a finite $S$ -module that is also finitely generated as an $R$ -module. Assume:

the closed fibre $M / m M$ is flat over $S / m S$ ;
the multiplication map $m \otimes_{R} M \to M$ is injective.

Then $M$ is flat over $R$ .

Facts & Assumptions

Given: A local map of Noetherian local rings $R \to S$ with maximal ideal $m$ , and a finite $S$ -module $M$ that is finitely generated as an $R$ -module and satisfies the two hypotheses.

[L1]

The ideal-form local criterion applies once one knows that $M / m M$ is flat over $R / m$ and that

m \otimes_{R} M \to M

is injective (For an $R$ -finite module over a local map, flatness modulo $I$ and injectivity of $I \otimes M \to M$ imply flatness).

Proof

technique · direct

1.1

The ring $R / m$ is a field. Since $M / m M$ is flat over the $S / m S$ -algebra in hypothesis 1, it is in particular a vector space over $R / m$ , hence flat over $R / m$ .

givenalgebra

1.2

Hypothesis 2 is exactly the injectivity condition required in [L1] for the ideal $I = m$ . Therefore [L1] applies and yields that $M$ is flat over $R$ .

L1given

2.1

This is the claimed closed-fibre criterion.

algebra∎

Depends on

For an $R$-finite module over a local map, flatness modulo $I$ and injectivity of $I \otimes M \to M$ imply flatness

Used by

Nothing in the library uses this result yet.

Dependency tree · two levels

7 results within two dependency steps of this one, each drawn at its shortest distance from it. An arrow runs from a result to what uses it, so the chart reads left to right and ends at this result, which carries a heavier outline. Every node is a link to that result. Click elsewhere on the chart to enlarge it.

Sources

Stacks Project, Lemma 10.99.15 (standard reference, not scraped)
Craig Huneke and Irena Swanson, Integral Closure, Chapter 2 (standard reference, not scraped)

For an RR-finite module over a local map, flatness on the closed fibre plus the multiplication-map condition implies flatness