ExampleConstruction: AI-generatedVerification: AI-adaptedPipeline-generatedprecheck passverified 2026-09-23 (gpt-6-sol)

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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.

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residue field infinite projective dimension singular

Example

Assume the Axiom of Choice. Let $k$ be a field. For $R = k [ε] / (ε^{2})$ , its residue field $k$ has an infinite minimal free resolution with one copy of $R$ in every degree and every positive differential multiplication by $ε$ . Consequently $β_{i}^{R} (k) = 1$ for all $i \geq 0$ and ${pd}_{R} k = \infty$ .

Facts & Assumptions

Given: The objects and hypotheses in the example. We work with the Axiom of Choice; cited dependent-choice and resolution-existence hypotheses are retained.

[F1]

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

[F2]

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 $i \geq 0$ .

Verification

1.1givenalgebra

The ring is local with maximal ideal $(ε)$ . For multiplication by $ε$ , the image and kernel both equal $(ε)$ : $ε (a + b ε) = a ε$ . The augmentation $R \to k$ has that same kernel. Thus the infinite augmented complex is exact in every degree and all positive matrix entries are in the maximal ideal.

2.1givenF2F1step 1.1∎

Under the stated AC premise, the rank formula gives $β_{i} (k) = 1$ in every degree. These nonzero Betti numbers are unbounded in degree, so the projective-dimension criterion gives infinity. A finite initial truncation would have a nonzero left kernel and is not a finite resolution.

Depends on

Used by

Nothing in the library uses this result yet.

Dependency tree · two levels

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Sources

Proposition 12.27 and Corollary 12.29, p.121 (standard reference, not scraped)