DefinitionDefinition: Literature-sourcedProof: Not applicablePipeline-generatedjudge pass (gpt-5.6-terra)audited 2026-08-31

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 shift of a chain complex

Definition

Let $C_{∙}$ be a chain complex and fix $k \in Z$ . The shift $C [k]_{∙}$ is the chain complex defined by $C [k]_{n} : = C_{n - k}, d_{n}^{C [k]} : = (- 1)^{k} d_{n - k}^{C} .$

Thus the underlying graded object is reindexed by $k$ , and the differential is twisted by the sign $(- 1)^{k}$ .

Depends on

Chain complex in an abelian category

The shift of a chain complex

Definition

Depends on

Dependency tree · 2 levels

Sources