LemmaStatement: Literature-sourcedProof: AI-adaptedPipeline-generatedprecheck passjudge pass (gpt-5.6-terra)audited 2026-09-07‡ rests on unproved material (inherited)

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.

‡ Rests on 3 statements not proved in this library, by way of the results it cites. This item cites no such statement directly; it depends on results that do. The unproved premises it inherits are ‡ Cohen's first model: an infinite Dedekind-finite set of reals, ‡ Sierpiński 1947: the generalised continuum hypothesis implies the Axiom of Choice and ‡ The continuum hypothesis and its generalisation are independent of ZFC. Each is recorded with a citation to the literature and is not established here, because the track that would prove it has not yet been developed in this library. Everything else in this proof is proved here.

Radical invariance of first nonzero Ext

Statement

Let $R$ be Noetherian, $M$ a finite $R$ -module, and $I, J$ ideals with $\sqrt{I} = \sqrt{J}$ . Then the first nonzero degrees of ${Ext}_{R}^{*} (R / I, M)$ and ${Ext}_{R}^{*} (R / J, M)$ are equal, with simultaneous value $\infty$ if both families vanish.

Facts & Assumptions

Given: The data in the statement.

Proof

technique · direct

1.1

Noetherianity gives $a, b > 0$ with $I^{a} \subseteq J$ and $J^{b} \subseteq I$ . The standard finite-filtration Ext devissage says that vanishing of ${Ext}_{R}^{i} (R / I, M)$ for $i < d$ is equivalent to the same vanishing for every finite module annihilated by a power of $I$ .

givenalgebra

2.1

Apply this first to $R / J$ , annihilated by $I^{a}$ , and then symmetrically to $R / I$ . The two Ext families vanish through the same initial range, including the all-vanishing case.

step 1.1∎

Depends on

Depth as the first nonzero Ext degree

Used by

Depth depends only on the radical of the ideal Corollary

Dependency tree · two levels

3 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

Depth and Cohen--Macaulay modules source treatment (standard reference, not scraped)