Strong convergence of a spectral sequence

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 ${\bigvee }_p{F}_p{H}_n=H_n,{\bigwedge }_p{F}_p{H}_n=0,$ as in Exhaustive separated bounded and finite filtration, and completeness means that the canonical map 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 $p^{\mathrm{\prime }}\le p$ is the quotient $H_n\mathrm{/}{F}_{p^{\mathrm{\prime }}}{H}_n\to H_n\mathrm{/}{F}_p{H}_n$. 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 $p\le a$ 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 and graded pieces $F^p\mathrm{/}{F}^{p+1}$. 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.