Convergence of a spectral sequence ($E_{\infty }^{p,q}\cong F^p{H}^{p+q}/F^{p+1}{H}^{p+q}$)

The stable page identifies the associated graded module of a filtration of the abutment. For a finite filtration this is an exhaustive, separated finite extension description. It does not itself determine a canonical splitting. Unbounded filtrations require further convergence hypotheses.

A finite filtration of a cochain complex produces a convergent spectral sequence. The induced abutment filtration is the image of $H^n(F^{p}C)\to H^n(C)$. Finiteness in each degree suffices; a bounded range of cochain degrees is not required.