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.

Associated graded object of a filtered object

Definition

For an increasing filtration, the associated graded object is the indexed family ${gr}_{p}^{F} A = coker (F_{p - 1} A ↪ F_{p} A) = F_{p} A / F_{p - 1} A (p \in Z) .$ For a decreasing filtration it is ${gr}_{F}^{p} A = F^{p} A / F^{p + 1} A$ . Here the inclusions come from Increasing and decreasing filtration of an object, and quotient means The quotient of an object by a subobject. This definition uses a family; it assumes no infinite direct sum. With $F^{p} = F_{- p}$ , the decreasing piece at $p$ is the increasing piece at $- p$ .

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.

Associated graded object of a filtered object

Definition

Source notes

Depends on

Dependency tree · 2 levels

Sources