DefinitionDefinition: AI-adaptedProof: Not applicablePipeline-generatedjudge pass (gpt-5.6-terra)audited 2026-09-12

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.

Filtered chain map

Definition

A filtered chain map $f : (C, d, F) \to (D, \partial, F)$ is a chain map satisfying $\partial_{n} f_{n} = f_{n - 1} d_{n}$ such that $f_{n}$ restricts to $F_{p} C_{n} \to F_{p} D_{n}$ for every $p, n \in Z$ . Here filtered complex means Filtered chain complex and chain map means Chain map. The restrictions are unique. The identity preserves each piece; if $f$ and $g$ preserve pieces, then $(g f) (F_{p} C_{n}) \subseteq g (F_{p} D_{n}) \subseteq F_{p} E_{n}$ , so their composite does too.

Source notes

Weibel, Chapter 5, §5.2 pp.122–125 and Definitions 5.4.2–4 pp.132–133; Sharifi, §4.1 pp.87–89 and Definition 4.2.1 p.90.

Filtered chain map

Definition

Source notes

Depends on

Dependency tree · 2 levels

Sources