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.

Increasing and decreasing filtration of an object

Definition

Let $A$ be an object of an abelian category. An increasing filtration is a specified family of subobjects $F_{p} A \subseteq A$ , $p \in Z$ , with $F_{p} A \subseteq F_{p + 1} A$ . A decreasing filtration satisfies $F^{p + 1} A \subseteq F^{p} A$ . The translation on the same object is $F^{p} A = F_{- p} A$ ; translating back gives $F_{p} A = F^{- p} A$ . A filtration-preserving morphism $f : A \to B$ factors on each piece as $F_{p} A \to F_{p} B ↪ B$ (or with upper indices). The restrictions are unique because the inclusions are monic. The ambient setting is Abelian category.

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.

Depends on

Abelian category

Increasing and decreasing filtration of an object

Definition

Source notes

Depends on

Dependency tree · 2 levels

Sources