LemmaStatement: Literature-sourcedProof: AI-adaptedPipeline-generatedprecheck passaudited 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.

Maximal regular sequences have a common Ext length

Statement

Let $R$ be Noetherian, let $M \neq 0$ be finite, and let $I$ lie in the Jacobson radical with $I M \neq M$ . Every maximal $M$ -regular sequence in $I$ has length $\min {i \geq 0 : {Ext}_{R}^{i} (R / I, M) \neq 0},$ so all such sequences have the same length.

Facts & Assumptions

Given: The data in the statement and a maximal regular sequence $x$ of length $r$ .

Proof

technique · direct

1.1

Put $Q = M / (x) M$ . Regularity makes $Q \neq 0$ , and Nakayama gives $I Q \neq Q$ . Maximality and the stopping criterion yield $(0 :_{Q} I) \neq 0$ , hence ${Ext}_{R}^{0} (R / I, Q) \neq 0$ .

given

2.1

Iterating the Ext shift backwards along the $r$ quotients shows that ${Ext}_{R}^{i} (R / I, M) = 0$ for $i < r$ and is nonzero for $i = r$ . Therefore the displayed minimum is $r$ , independently of the sequence.

step 1.1algebra∎

Depends on

Used by

Depth as the first nonzero Ext degree Corollary
Depth equals the maximal regular-sequence length Theorem

Dependency tree · two levels

7 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)