CorollaryStatement: Literature-sourcedProof: AI-adaptedPipeline-generatedprecheck passjudge 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.

Empty Koszul Complex Is The Coefficient Module

Statement

For the empty sequence, $K (\emptyset; M)$ is $M$ in degree $0$ and $0$ in every other degree.

Facts & Assumptions

Given: The rings, finite sequences, modules, and local hypotheses stated in the claim. The declared prerequisites used here are Koszul Complex Of A Sequence With Coefficients.

Proof

technique · direct

1.1

For the zero free module, only $⋀^{0} 0 = R$ is nonzero.

givenalgebra

2.1

Tensoring with $M$ gives $M$ in degree zero and zero elsewhere, with zero differential.

step 1.1algebra∎

Depends on

Koszul Complex Of A Sequence With Coefficients

Used by

Empty And Unit Koszul Boundaries Example
Regular Sequences Give Acyclic Koszul Complexes Theorem

Dependency tree · two levels

4 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)