LemmaStatement: 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.
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auslander buchsbaum syzygy projective dimension

Statement

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 $pd K = n - 1$ .

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]

projective dimension from last nonzero betti number: For a nonzero finite module $M$ over a nonzero Noetherian local ring, ${pd}_{R} M = \sup {i \geq 0 : β_{i}^{R} (M) \neq 0}$ , allowing infinity. For each integer $q \geq 0$ , ${pd}_{R} M \leq q$ if and only if ${Tor}_{q + 1}^{R} (k, M) = 0$ .

Proof

1.1

The minimal resolution of $M$ has last nonzero term $F_{n}$ by the Betti criterion. Truncating it gives a minimal resolution $\dots \to F_{2} \to F_{1} \to K \to 0$ . If $K = 0$ , the initial presentation would make $M$ free, contrary to $n > 0$ .

2.1

The truncated resolution has last nonzero term $F_{n}$ in degree $n - 1$ , so the same criterion gives $pd K = n - 1$ . For $n = 1$ this says that $K$ is a nonzero finite free module.

F1step 1.1∎

Depends on

projective dimension from last nonzero betti number

Used by

auslander buchsbaum first syzygy Example
auslander buchsbaum first syzygy depth Lemma
residue field splits off reduced maximal ideal Lemma
auslander buchsbaum formula Theorem

Dependency tree · two levels

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Sources

Theorem 12.31 proof, Case 3, p.122 (standard reference, not scraped)