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

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auslander buchsbaum formula

Statement

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.

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]

auslander buchsbaum base case free module: If a nonzero finite module $M$ over a nonzero Noetherian local ring $R$ has projective dimension zero, then it is finite free of positive rank and ${depth}_{R} M = depth R$ .

[F2]

auslander buchsbaum projective dimension one: If $M$ is a nonzero finite module of projective dimension one over a nonzero Noetherian local ring $R$ , then $depth R \geq 1$ and $depth M = depth R - 1$ .

[F3]

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 $pd K = n - 1$ .

[F4]

auslander buchsbaum first syzygy depth: In a minimal presentation $0 \to K \to F \to M \to 0$ of a nonzero finite module over a nonzero Noetherian local ring, let $n = pd M \geq 2$ be finite. If $depth K = depth R - (n - 1)$ , then $depth M = depth K - 1$ .

Proof

1.1

Induct on $n = pd M$ . For $n = 0$ the module is nonzero finite free and has the ring depth. For $n = 1$ the separate minimal-matrix argument proves the formula.

F1F2

2.1

For $n \geq 2$ , take the first syzygy $K$ in a minimal presentation. It is nonzero finite with projective dimension $n - 1$ , so the inductive assertion gives $depth K = depth R - (n - 1)$ . The conditional syzygy-depth lemma then gives $depth M = depth R - n$ . This completes the induction.

F3F4step 1.1

3.1

If $depth M = depth R$ , the formula forces projective dimension zero, and the base-case theorem gives freeness. The nonzero and finite-projective-dimension hypotheses are retained throughout.

F1step 2.1algebra∎

Depends on

Used by

Dependency tree · two levels

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Sources

Theorem 1.53, pp.24–25; Mustata 12.31 (standard reference, not scraped)