ExampleConstruction: AI-generatedVerification: AI-adaptedPipeline-generatedverified 2026-09-23 (gpt-6-astra)

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.

auslander buchsbaum first syzygy

Example

Assume the Axiom of Choice (The Axiom of Choice).

For $R = k [x, y]_{(x, y)}$ with maximal ideal $m$ , ${pd}_{R} k = 2$ , ${depth}_{R} k = 0$ , and its first syzygy satisfies ${pd}_{R} m = 1$ and ${depth}_{R} m = 1$ .

Facts & Assumptions

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

[F1]

auslander buchsbaum syzygy projective dimension: Let $0 \to K \to F_{0} \to M \to 0$ be the initial minimal presentation of a nonzero finite module over a nonzero Noetherian local ring. If $0 < n = pd M < \infty$ , then $K \neq 0$ and $pd K = n - 1$ .

[F2]

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.

[F3]

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 $i > d$ .

[F4]

localisation and polynomial extension of regular rings: Localizations and finite polynomial extensions of a commutative regular Noetherian ring are regular. Regularity can equivalently be tested at maximal ideals. For every nonzero such ring, $gldim R = \dim R$ , allowing infinity. More generally, for a finite module over any commutative Noetherian ring, projective dimension is the supremum of its prime-local projective dimensions. Dedekind domains and their finite polynomial extensions are regular.

Verification

1.1F4F3algebra

The ring is regular local of dimension two, by polynomial regularity and the coordinate chain and generator count. The residue-field computation gives projective dimension two. Its depth is zero because every maximal-ideal element kills the nonzero module $k$ . The presentation $0 \to m \to R \to k \to 0$ is minimal.

2.1F1F2step 1.1algebra∎

The syzygy theorem gives $pd m = 1$ . The ring depth is two, since $x, y$ is a regular sequence and depth is bounded by dimension. Auslander–Buchsbaum gives $depth m = 2 - 1 = 1$ . Concretely its minimal resolution is $0 \to R \overset{c \mapsto (- y c, x c)}{\to} R^{2} \to m \to 0$ : reducing a relation modulo $x$ shows $b = x c$ , and then cancellation gives $a = - y c$ .

Depends on

Used by

Nothing in the library uses this result yet.

Dependency tree · two levels

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Sources

Theorem 12.31 and Theorem 12.33, pp.121–123 (standard reference, not scraped)