DefinitionDefinition: Literature-sourcedProof: Not applicablePipeline-generatedjudge pass (gpt-5.6-terra)audited 2026-09-06

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.

Regular Sequence On A Module

Definition

Let $R$ be a commutative unital ring, let $M$ be an $R$ -module, and let $x = (x_{1}, \dots, x_{n})$ be a finite ordered sequence in $R$ . The sequence is $M$ -regular when $M / (x_{1}, \dots, x_{i - 1}) M \neq 0$ and multiplication by $x_{i}$ is injective on it for every $i$ , and $M / (x) M \neq 0$ .

Depends on

Used by

Empty And Unit Koszul Boundaries Example
Nonpermutable Regular Sequence Example
Local Koszul Acyclicity Inductive Converse Lemma
Local Koszul H One Detects First Regularity Failure Lemma
Positive Powers Of A Regular Sequence Remain Regular Lemma
Regular One Element Koszul Acyclicity Lemma
Regular Sequence First Element Boundary Lemma
Regular Sequence Permutation Adjacent Swap Lemma
Regular Sequence Tail On Quotient Lemma
Localisation And Faithfully Flat Base Change Of Regular Sequences Theorem
Regular Sequences Give Acyclic Koszul Complexes 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

The Stacks Project, Koszul complexes and regular sequences (standard reference, not scraped)