DefinitionDefinition: Literature-sourcedProof: Not applicablePipeline-generatedjudge 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 direct-sum total complex on finite diagonals

Definition

For the first-quadrant Hom bicomplex $K$ , put ${Tot}^{n} K = ⨁_{p + q = n} K^{p, q}, D ∣_{K^{p, q}} = h^{p, q} + (- 1)^{p} v^{p, q} .$ Each diagonal has precisely $n + 1$ possible bidegrees, so this direct sum is finite. Consequently there is no choice here between direct-sum and product totalisations.

Depends on

Dependency tree · 2 levels

Sources

Charles A. Weibel, An Introduction to Homological Algebra, Chapter 2