DefinitionDefinition: Literature-sourcedProof: Not applicablePipeline-generatedjudge pass (gpt-5.6-terra)audited 2026-09-07

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.

betti numbers of a finite local module

Definition

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.

Depends on

Used by

betti number is rank in minimal resolution Corollary

Dependency tree · two levels

10 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

Unnumbered Betti-number definition after Lemma 1.50, p.24 (standard reference, not scraped)