TheoremStatement: Literature-sourcedProof: AI-adaptedprecheck passaudited 2026-09-01

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Literature-sourced: the exact statement appears in a cited source; only wording and notation differ.
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Every flat ring map satisfies going down

Statement

Assume the Axiom of Choice for the prime-ideal lifting step.

Let $f : R \to S$ be a flat homomorphism of commutative rings. Then $f$ satisfies going down: whenever $p_{1} \subseteq p_{2} \subset R, q_{2} \subset S, q_{2} \cap R = p_{2},$ there exists a prime ideal $q_{1} \subseteq q_{2}$ with $q_{1} \cap R = p_{1} .$

Facts & Assumptions

Given: A flat ring map $f : R \to S$ and primes $p_{1} \subseteq p_{2} \subset R$ , $q_{2} \subset S$ with $q_{2} \cap R = p_{2}$ .

[L1]

Localizations of flat maps are flat (Every localization is flat, and localizing a flat module preserves flatness).

[L2]

A flat local map is faithfully flat on the localized spectra criterion (A flat ring map is faithfully flat exactly when it detects proper ideals and is surjective on spectra).

Proof

technique · direct

1.1L1given

Localize at $p_{2}$ and then at $q_{2}$ . We obtain a flat local homomorphism $R_{p_{2}} \to S_{q_{2}}$ by [L1].

1.2L2algebra

Because this localized map is local and flat, [L2] makes it faithfully flat. Applying the spectral characterization of faithful flatness to the prime ideal $p_{1} R_{p_{2}}$ of the source produces a prime $q_{1} S_{q_{2}} \subseteq S_{q_{2}}$ lying over it. Contracting back to $S$ gives a prime $q_{1} \subseteq q_{2}$ with $q_{1} \cap R = p_{1} .$

2.1algebra∎

Therefore flat ring maps satisfy going down.

Depends on

Used by

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, Lemmas 10.39.17 and 10.39.19 (standard reference, not scraped)
J. S. Milne, A Primer of Commutative Algebra, Proposition 11.20 (standard reference, not scraped)