TheoremStatement: Literature-sourcedProof: AI-adaptedPipeline-generatedjudge pass (gpt-5.6-terra)audited 2026-09-07‡ rests on unproved material (inherited)

How statement and proof provenance work

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.

These labels describe origin, not correctness: citations and verification chips remain separate evidence.

‡ Rests on 3 statements not proved in this library, by way of the results it cites. This item cites no such statement directly; it depends on results that do. The unproved premises it inherits are ‡ Cohen's first model: an infinite Dedekind-finite set of reals, ‡ Sierpiński 1947: the generalised continuum hypothesis implies the Axiom of Choice and ‡ The continuum hypothesis and its generalisation are independent of ZFC. Each is recorded with a citation to the literature and is not established here, because the track that would prove it has not yet been developed in this library. Everything else in this proof is proved here.

auslander buchsbaum serre regularity criterion

Statement

For a nonzero Noetherian local ring $(R, m, k)$ the following are equivalent: $R$ is regular; ${pd}_{R} k < \infty$ ; $gldim R < \infty$ ; and every finite $R$ -module has finite projective dimension. When these hold, $gldim R = {pd}_{R} k = \dim R$ . A nonzero finite module over regular local $R$ is maximal Cohen–Macaulay (depth $\dim R$ ) if and only if it is free.

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]

finite residue field projective dimension forces depth equals dimension: If the residue field of a nonzero Noetherian local ring $R$ has finite projective dimension, then $R$ is regular and $depth R = \dim R = edim R$ .

[F2]

regular local residue field projective dimension dimension: For a regular local ring $(R, m, k)$ of dimension $d$ , ${pd}_{R} k = d$ and $β_{i}^{R} (k) = (\binom{d}{i})$ for $0 \leq i \leq d$ , with $β_{i}^{R} (k) = 0$ for $i > d$ .

[F3]

local global dimension equals residue field projective dimension: For a nonzero Noetherian local ring $(R, m, k)$ , $gldim R = {pd}_{R} k$ , allowing infinity. If this common value is $n < \infty$ , every $R$ -module has projective dimension at most $n$ .

[F4]

auslander buchsbaum formula: For a nonzero finite module $M$ of finite projective dimension over a nonzero Noetherian local ring $R$ , ${pd}_{R} M + {depth}_{R} M = depth R$ . Consequently such an $M$ with $depth M = depth R$ is free.

[F5]

regular local rings are domains and cohen macaulay: A regular local ring $R$ of dimension $d$ is a domain and Cohen–Macaulay. For every regular system $(x_{1}, \dots, x_{d})$ , the tuple is $R$ -regular and $R / (x_{1}, \dots, x_{c})$ is regular local of dimension $d - c$ for all $0 \leq c \leq d$ .

[F6]

Assuming the Axiom of Choice, Nakayama's lemma: Assume the Axiom of Choice. Let $R$ be a commutative ring, let $I ⊴ R$ satisfy $I \subseteq J (R)$ , and let $M$ be a finitely generated left $R$ -module. If $I M = M$ , then $M = 0$ .

Proof

1.1

Regularity gives $pd k = \dim R$ through the Koszul computation. Residue-field projective dimension equals global dimension, so this also bounds every module. Conversely finite projective dimension for every finite module applies to $k$ , and finite projective dimension for $k$ forces regularity. These implications prove the four-way equivalence and the numerical equalities, also for dimension zero.

F2F3F1

2.1

Over a regular local ring every finite module has finite projective dimension and $depth R = \dim R$ . Auslander–Buchsbaum therefore makes depth $\dim R$ equivalent to projective dimension zero for a nonzero finite module, hence equivalent to freeness. Conversely a nonzero finite free module has the ring depth.

F4F5step 1.1

3.1

The useful freeness-lifting argument can also be seen directly. If $x \in m$ is injective on a finite $M$ and $M / x M$ is free over $R / (x)$ , lift a basis to a surjection $F \to M$ by Nakayama, with finite kernel $K$ . A relation has coefficients divisible by $x$ , so it is $x v$ ; injectivity on $M$ implies $v \in K$ . Thus $K = x K$ , and Nakayama gives $K = 0$ . This includes a zero quotient basis, when $M = 0$ .

F6algebra∎

Depends on

Used by

localisations of regular local rings are regular Corollary
Cohen–Macaulayness over a finite regular local base Example
flat local ascent of regularity Lemma
localisation and polynomial extension of regular rings Theorem

Dependency tree · two levels

30 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

Theorem 12.33 and Corollary 12.30, pp.121–123 (standard reference, not scraped)