DefinitionDefinition: Literature-sourcedProof: Not applicablePipeline-generatedaudited 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.

Koszul Regular And H One Regular Sequences

Definition

Let $R$ be a commutative unital ring, let $M$ be an $R$ -module, and let $x$ be a finite ordered sequence in $R$ . Call $x$ $M$ -Koszul-regular when $H_{i} (K (x; M)) = 0$ for every $i > 0$ , and $M$ - $H_{1}$ -regular when $H_{1} (K (x; M)) = 0$ ; ordinary regularity is the preceding ordered definition.

Depends on

Used by

Dependency tree · two levels

8 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)