Double cochain complex ($d_h:D^{p,q}\to D^{p+1,q},d_v:D^{p,q}\to D^{p,q+1}$)

Two degree-raising differentials square to zero and anticommute. If an initial convention uses commuting differentials, introduce a sign in one of them before forming the total cochain complex.

Filtering the total cochain complex by each index yields two spectral sequences. One first takes vertical cohomology and the other horizontal cohomology; boundedness ensures both converge to total cohomology.

The total differential is $d_h+d_v$ for anticommuting differentials. For a double cochain complex bounded in both indices, both index filtrations are finite and have the same total cohomology as abutment.