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.

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AI-generated: a genuinely novel statement formulated by AI, with no source for the claim itself.

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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.

Depends on

Used by

Isomorphic associated graded objects need not give isomorphic filtered objects Counterexample
Abutment to a filtered object Definition
Associated graded quotients are well defined subquotients Lemma

Dependency tree · two levels

5 results within two dependency steps of this one, each drawn at its shortest distance from it. An arrow runs from a result to what uses it, so the chart reads left to right and ends at this result, which carries a heavier outline. Every node is a link to that result. Click elsewhere on the chart to enlarge it.

Sources

Charles A. Weibel, An Introduction to Homological Algebra, Chapter 5 (standard reference, not scraped)
Romyar Sharifi, Homological Algebra (standard reference, not scraped)