Spectral sequence ($(E_r^{p,q},d_r)$)

A succession of bigraded pages with $E_{r+1}=H(E_r,d_r)$. In the cohomological convention, the differential has bidegree $(r,1-r)$. A filtered complex produces such successive approximations to its cohomology.

For a bounded complex $P$ of projective right modules and a left module $M$ of finite projective dimension, a finite projective resolution of $M$ yields a bounded double cochain complex. Its two spectral sequences give the displayed abutment. Over a noncommutative ring, the handedness of the two tensor factors is essential.

If only the columns $p=-1,0$ are nonzero, a cohomological differential of bidegree $(r,1-r)$ has no possible nonzero source-target pair for $r\ge 2$. Projective boundaries in a complex of projectives give this condition because every cohomology module has projective dimension at most one.

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.