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

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Literature-sourced: the exact statement appears in a cited source; only wording and notation differ.
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betti number is rank in minimal resolution

Statement

For every minimal degreewise finite free resolution $F_{∙} \to M$ of a finite module over a nonzero Noetherian local ring, $β_{i}^{R} (M) = {rank}_{R} F_{i}$ for all $i \geq 0$ .

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]

betti numbers of a finite local module: For a finite module $M$ over a nonzero Noetherian local ring $(R, m, k)$ and an integer $i \geq 0$ , its Betti number is $β_{i}^{R} (M) = \dim_{k} {Tor}_{i}^{R} (k, M)$ . The action factors through $k$ , and a degreewise finite free resolution makes this dimension finite. Tor is resolution-independent. This extends the Koszul rank notation: whenever a minimal Koszul resolution exists, the rank formula identifies these numbers with its Koszul Betti numbers. For $M = 0$ all Betti numbers are zero.

[F2]

minimal free resolution reduces to zero differential: If $F_{∙} \to M$ is a minimal degreewise finite free resolution over $(R, m, k)$ , every differential of the unaugmented complex $k \otimes_{R} F_{∙}$ is zero.

Proof

1.1

The residue complex has zero differentials and computes ${Tor}_{i}^{R} (k, M)$ , so this Tor group is $k \otimes_{R} F_{i}$ .

2.1

Its vector-space dimension equals the finite free rank of $F_{i}$ . By definition this is $β_{i}^{R} (M)$ . This holds in degree zero, in all higher degrees, and for zero terms, including the zero module.

F1step 1.1algebra∎

Depends on

Used by

Dependency tree · two levels

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Sources

Proposition 12.27, pp.120–121 (standard reference, not scraped)