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

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

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.

technique · direct

1.1givenalgebra

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

2.1step 1.1algebra∎

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

Depends on