LemmaStatement: Literature-sourcedProof: AI-adaptedPipeline-generatedprecheck passaudited 2026-09-07

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Literature-sourced: the exact statement appears in a cited source; only wording and notation differ.
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The first nonzero Ext shifts across a regular element

Statement

Let $R$ be Noetherian, $M$ a finite $R$ -module, $I$ an ideal, and $x \in I$ an $M$ -regular element with $M / x M \neq 0$ . If $d = \min {i : {Ext}_{R}^{i} (R / I, M) \neq 0} < \infty,$ then the first nonzero Ext degree for $M / x M$ is $d - 1$ .

Facts & Assumptions

Given: The data and finite integer $d$ in the statement.

Proof

technique · direct

1.1givenalgebra

Apply ${Ext}_{R} (R / I, -)$ to $0 \to M \overset{x}{\to} M \to M / x M \to 0$ . Multiplication by $x$ on every ${Ext}_{R}^{i} (R / I, M)$ is zero because $x$ annihilates $R / I$ . The long exact sequence therefore gives $0 \to {Ext}_{R}^{i} (R / I, M) \to {Ext}_{R}^{i} (R / I, M / x M) \to {Ext}_{R}^{i + 1} (R / I, M) \to 0.$

2.1step 1.1∎

For $i < d - 1$ both outer groups vanish, while at $i = d - 1$ the right-hand group is nonzero and the left-hand group is zero. This proves the shift.

Depends on

Used by

Nothing in the library uses this result yet.

Dependency tree · two levels

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Sources

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