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 complex

Definition

A filtered chain complex $(C, d, F)$ in an abelian category is a chain complex with increasing filtrations $F_{p} C_{n} \subseteq C_{n}$ for all $n$ , whose pieces form subcomplexes. Explicitly $d_{n}$ factors as $F_{p} C_{n} \to F_{p} C_{n - 1}$ on each piece. These restrictions square to zero since their composites with the monic inclusion into $C_{n - 2}$ are $d_{n - 1} d_{n} = 0$ . The conventions are Increasing and decreasing filtration of an object and Subcomplex. Thus, in modules, the condition is exactly $d_{n} (F_{p} C_{n}) \subseteq F_{p} C_{n - 1}$ ; a factorization implies the inclusion, and the inclusion defines the restricted function. No finite-bound hypothesis is implicit.

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 complex

Definition

Source notes

Depends on

Dependency tree · 2 levels

Sources