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.

Induced filtration on homology

Definition

For a filtered chain complex, the induced filtration on homology is $F_{p} H_{n} (C) = im (H_{n} (F_{p} C) ⟶ H_{n} (C)) .$ The arrow is induced by the inclusion chain map from Filtered chain complex, using A chain map induces a well-defined map on homology. Since $F_{p - 1} C \to C$ factors through $F_{p} C$ , functoriality in Homology respects identities and composition shows that its homology image is contained in $F_{p} H_{n} (C)$ . Thus this is an increasing family of subobjects of $H_{n} (C)$ . The domain $H_{n} (F_{p} C)$ of the arrow need not embed into $H_{n} (C)$ .

Induced filtration on homology

Definition

Source notes

Depends on

Dependency tree · 2 levels

Sources