CorollaryStatement: Literature-sourcedProof: AI-adaptedPipeline-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.

regular local residue field projective dimension dimension

Statement

For a regular local ring $(R, m, k)$ of dimension $d$ , ${pd}_{R} k = d$ and $β_{i}^{R} (k) = (\binom{d}{i})$ for $0 \leq i \leq d$ , with $β_{i}^{R} (k) = 0$ for $i > d$ .

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]

regular local residue field koszul resolution: For a regular local ring $(R, m, k)$ of dimension $d$ , the Koszul complex on any regular system of parameters is a minimal free resolution of $k$ of length $d$ .

[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]

betti number is rank in minimal resolution: 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 $i \geq 0$ .

[F4]

Complete Intersection Betti Numbers Binomial: For a length- $n$ regular sequence in the maximal ideal of a local ring, the minimal Koszul resolution has $β_{i}^{K} = (\binom{n}{i})$ for $0 \leq i \leq n$ and $0$ otherwise.

Proof

1.1

The minimal Koszul resolution has degree- $i$ rank $(\binom{d}{i})$ , and is zero above $d$ . The Koszul rank formula and the general minimal-resolution rank formula identify these with the stated Betti numbers.

F1F4F3

2.1

The top rank $(\binom{d}{d}) = 1$ is nonzero, so the projective-dimension criterion gives exactly $d$ , not merely an upper bound. For $d = 0$ the sole rank is $β_{0} (k) = 1$ .

F2step 1.1algebra∎

Depends on

Used by

auslander buchsbaum first syzygy Example
betti numbers residue field regular ring Example
auslander buchsbaum serre regularity criterion Theorem

Dependency tree · two levels

19 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

12.27 and 12.33, pp.121–123 (standard reference, not scraped)