CorollaryStatement: Literature-sourcedProof: AI-adaptedPipeline-generatedprecheck passverified 2026-09-23 (gpt-6-sol)

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regular local residue field projective dimension dimension

Statement

Assume the Axiom of Choice. 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

1.1F1F4F3

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.

2.1givenF2step 1.1algebra∎

Under the stated AC premise, [F2] applies to the top rank $(\binom{d}{d}) = 1$ and gives projective dimension exactly $d$ , not merely an upper bound. For $d = 0$ the sole rank is $β_{0} (k) = 1$ .

Depends on

Dependency tree · 2 levels

Sources

12.27 and 12.33, pp.121–123