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

Used by

The Hom double complex in low bidegrees Example
FALSE: double-complex totalisation is unambiguous with infinite diagonals False statement
Acyclic assembly by exact columns Lemma
Acyclic assembly by exact rows Lemma
Projective and injective constructions of Ext agree for supplied resolutions Theorem
The bar cochain complex computes group cohomology 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

Charles A. Weibel, An Introduction to Homological Algebra, Chapter 2 (standard reference, not scraped)