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 rows of the tensor double complex are exact

If $Q_{∙} ↠ N$ and $P_{∙} ↠ M$ are projective resolutions, every augmented row $Q_{p} \otimes_{R} P_{∙} \to Q_{p} \otimes_{R} M$ is exact.

Given: a fixed projective right module $Q_{p}$ and the augmented resolution $P_{∙} \to M$ .

1.1given

The module $Q_{p}$ is flat as a right module, so $Q_{p} \otimes_{R} -$ preserves the exact augmented complex $P_{∙} \to M$ .

2.1step 1.1algebra

Its degree- $q$ terms are exactly $Q_{p} \otimes_{R} P_{q}$ , and its augmentation is $Q_{p} \otimes_{R} P_{0} \to Q_{p} \otimes_{R} M$ .

3.1step 2.1algebra∎

Thus the row indexed by $p$ is exact, including the augmentation; this is the claimed augmented-row condition.

Depends on