LemmaStatement: 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.

The augmented fixed-q rows of the tensor double complex are exact

For the same two projective resolutions, every augmented fixed- $q$ row $Q_{∙} \otimes_{R} P_{q} \to N \otimes_{R} P_{q}$ is exact.

Given: a fixed projective left module $P_{q}$ and the augmented resolution $Q_{∙} \to N$ .

1.1given

The module $P_{q}$ is flat as a left module, so $- \otimes_{R} P_{q}$ preserves exactness of $Q_{∙} \to N$ .

2.1step 1.1algebra

The resulting degree- $p$ terms are $Q_{p} \otimes_{R} P_{q}$ , with augmentation $Q_{0} \otimes_{R} P_{q} \to N \otimes_{R} P_{q}$ .

3.1step 2.1algebra∎

Hence every augmented fixed- $q$ row is exact, on the correct right-module/left-module tensor convention.

Depends on