DefinitionDefinition: AI-adaptedProof: Not applicablePipeline-generatedjudge pass (gpt-5.6-terra)

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.

Strong convergence of a spectral sequence

Definition

A spectral sequence converges strongly to $(H_{n}, F)$ on this page if it converges weakly with specified identifications as in Weak convergence of a spectral sequence, is two-sided regular as in Regular spectral sequence, and for each $n$ its target filtration is exhaustive, separated and complete. Here exhaustiveness and separatedness mean $⋁_{p} F_{p} H_{n} = H_{n}, ⋀_{p} F_{p} H_{n} = 0,$ as in Exhaustive separated bounded and finite filtration, and completeness means that the canonical map $H_{n} ⟶ {\underset{\leftarrow}{l i m}}_{p \to - \infty} H_{n} / F_{p} H_{n}$ is an isomorphism. All indicated subobject meets, joins and inverse limits must exist. The limit has the universal-property meaning of Limits and colimits as terminal cones and initial cocones, with existence and uniqueness in their universal properties: the transition for $p^{'} \leq p$ is the quotient $H_{n} / F_{p^{'}} H_{n} \to H_{n} / F_{p} H_{n}$ . In modules the limit is the module of compatible residue classes. All these conditions are required; the weak-convergence identifications remain part of the data.

A finite filtration is complete: if $F_{a} H_{n} = 0$ , every quotient with $p \leq a$ is canonically $H_{n}$ , with identity transitions. A cone is determined by its component at any such index, and its components at larger indices are its quotient maps. Thus $H_{n}$ itself is the inverse limit, even if the ambient category does not admit arbitrary inverse limits. The same finite lower endpoint gives separatedness, and a finite upper endpoint $F_{b} H_{n} = H_{n}$ gives exhaustiveness. This includes $H_{n} = 0$ and repeated filtration terms.

For decreasing cohomological filtrations use $H^{n} \to {\underset{\leftarrow}{l i m}}_{p \to + \infty} H^{n} / F^{p} H^{n}$ and graded pieces $F^{p} / F^{p + 1}$ . No splitting of the filtration and no choice axiom is included. The two-sided regularity convention is stronger than the outgoing-only meaning of regularity in Stacks, Definition 12.24.9; a source criterion must be checked against every condition above.

Depends on

Used by

Quasi isomorphism criterion from a filtered map Corollary
Exhaustive filtration implies separated and complete filtration False statement
Finite and complete filtered isomorphism lifting Lemma
Failure of separatedness or completeness can destroy the claimed abutment Proposition
A first quadrant filtered complex spectral sequence converges to filtered homology Theorem
Complete exhaustive filtered complex convergence criterion Theorem
Spectral sequence comparison theorem Theorem

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

Weibel, An Introduction to Homological Algebra, Chapter 5 (standard reference, not scraped)
The Stacks Project, Homological Algebra (standard reference, not scraped)