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 Hom double complex of projective and injective resolutions

Definition

For supplied resolutions $P_{∙} (M) \to M$ and $N \to I^{∙} (N)$ , define the first-quadrant bigraded object $K^{p, q} = {Hom}_{A} (P_{p} (M), I^{q} (N)) (p, q \geq 0) .$ Its horizontal and vertical maps are $h^{p, q} (f) = f \circ d_{P, p + 1}, v^{p, q} (f) = d_{I}^{q} \circ f .$ They have bidegrees $(1, 0)$ and $(0, 1)$ respectively. The first-quadrant restriction is part of the definition used below.

Depends on

Used by

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)