CorollaryStatement: Literature-sourcedProof: AI-adaptedPipeline-generatedjudge pass (gpt-5.6-terra)audited 2026-09-07

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The first chip identifies the source of the statement or construction; the second identifies the source of its local proof or verification.

Literature-sourced: the exact statement appears in a cited source; only wording and notation differ.
AI-adapted: a semantically identical restatement of literature-sourced material, modulo indexing, notation, and boundary cases adopted by the library.
AI-generated: a genuinely novel statement formulated by AI, with no source for the claim itself.

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serre normality criterion two directions

Statement

A commutative Noetherian domain is normal if and only if it satisfies $(R_{1})$ and $(S_{2})$ . Equivalently its integral closedness is characterized by these two conditions.

Facts & Assumptions

Given: The objects and hypotheses in the statement. We work with the Axiom of Choice; cited dependent-choice and resolution-existence hypotheses are retained.

[F1]

normal domain implies r one: Every commutative Noetherian integrally closed domain satisfies $(R_{1})$ .

[F2]

normal domain implies s two: Every commutative Noetherian integrally closed domain satisfies $(S_{2})$ .

[F3]

r one s two integral element membership: A commutative Noetherian $(S_{2})$ domain whose height-one localizations are DVRs is integrally closed.

[F4]

one dimensional regular local rings are dvrs: A nonzero Noetherian local ring of dimension one is regular if and only if it is a discrete valuation ring. Fields are excluded from the term DVR.

[F5]

A domain is integrally closed if and only if its prime localisations are, equivalently if and only if its maximal localisations are: Assume the Axiom of Choice. Let $A$ be a domain. Then the following are equivalent: 1. $A$ is integrally closed. 2. For every prime ideal $p$ of $A$ , the localisation $A_{p}$ is integrally closed. 3. For every maximal ideal $m$ of $A$ , the localisation $A_{m}$ is integrally closed.

Proof

1.1

For a domain, normality is equivalent to integral closedness by local normality. An integrally closed Noetherian domain satisfies $(R_{1})$ and $(S_{2})$ by the two normal-domain lemmas.

F5F1F2

2.1

Conversely, $(R_{1})$ makes every height-one localization one-dimensional regular local and hence a DVR. With $(S_{2})$ , the integral-element membership lemma makes $R$ integrally closed, and local normality makes it normal. Fields satisfy both conditions and are included.

F4F3F5∎

Depends on

Used by

serre normality criterion Theorem

Dependency tree · two levels

23 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

Proposition 8.41 and Lemma 8.40, pp.56–58 (standard reference, not scraped)