LemmaStatement: Literature-sourcedProof: AI-adaptedPipeline-generatedverified 2026-09-26 (gpt-6-sol)

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auslander buchsbaum projective dimension one

Statement

Assume the Axiom of Choice. If $M$ is a nonzero finite module of projective dimension one over a nonzero Noetherian local ring $R$ , then $depth R \geq 1$ and $depth M = depth R - 1$ .

Facts & Assumptions

Given: The objects and hypotheses in the statement, including the Axiom of Choice (The Axiom of Choice). Cited dependent-choice and resolution-existence hypotheses are retained.

[F1]

minimal free matrix induces zero on residue ext: For a nonzero commutative Noetherian local ring $(R, m, k)$ , let $α : R^{s} \to R^{t}$ be a map between finite free modules all of whose matrix entries lie in $m$ . Then ${Ext}_{R}^{i} (k, α) = 0$ for every $i \geq 0$ .

[F2]

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 ${Tor}_{q + 1}^{R} (k, M) = 0$ .

[F3]

Depth as the first nonzero Ext degree: Let $R$ be Noetherian, let $M$ be finite, and let $I$ lie in the Jacobson radical. Then ${depth}_{I} (M) = \inf {i \geq 0 : {Ext}_{R}^{i} (R / I, M) \neq 0},$ where the infimum of the empty set is $\infty$ .

[F4]

The long exact Ext sequence in the second variable: Assume the Axiom of Dependent Choice. Let $A$ be abelian with enough projectives and enough injectives, and fix supplied projective and injective resolution data on all its objects. For $0 \to N^{'} \to N \to N^{''} \to 0$ and every $M$ , there is a natural exact sequence $0 \to Hom (M, N^{'}) \to Hom (M, N) \to Hom (M, N^{''}) \overset{δ^{0}}{\to} {Ext}^{1} (M, N^{'}) \to {Ext}^{1} (M, N) \to \dots,$ where $δ^{q} : {Ext}^{q} (M, N^{''}) \to {Ext}^{q + 1} (M, N^{'})$ ; it is natural in the short exact sequence and contravariantly natural in $M$ .

[F5]

finite local modules admit minimal free resolutions: Every finite module over a nonzero Noetherian local ring has a degreewise finite minimal free resolution.

Proof

1.1F1F2F3F4F5

Choose the minimal resolution supplied by [F5]. Since ${pd}_{R} M = 1$ , [F2] says that its last nonzero term is $F_{1} ≅ R^{s}$ with $s > 0$ and $F_{i} = 0$ for $i \geq 2$ . Thus it gives a minimal exact sequence $0 \to R^{s} \overset{α}{\to} R^{t} \to M \to 0$ . Write $r = depth R$ . The map on every ${Ext}_{R}^{i} (k, -)$ induced by $α$ is zero. If $r = 0$ , the injection $Hom (k, R^{s}) \to Hom (k, R^{t})$ would be zero with nonzero source, impossible. Hence $r \geq 1$ .

2.1F4F3F1step 1.1∎

For $i < r - 1$ , the adjacent Ext terms for the free modules vanish, so ${Ext}_{R}^{i} (k, M) = 0$ . At $i = r - 1$ , exactness and the zero map in degree $r$ identify ${Ext}_{R}^{r - 1} (k, M)$ with ${Ext}_{R}^{r} (k, R^{s}) \neq 0$ . Thus the first nonzero Ext degree is $r - 1$ , proving the depth formula, also for $r = 1$ .

Depends on

Used by

auslander buchsbaum formula Theorem

Dependency tree · two levels

26 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 1.53, pp.24–25 (standard reference, not scraped)