A double complex has modules and differentials lowering either index, each squaring to zero and anticommuting with the other. The total chain complex has term and differential the sum of the two differentials. For the tensor product of two chain complexes, a sign on the second differential ensures anticommutation. If the double complex is in the first quadrant and one direction has homology only in degree zero, its total homology is computed by the surviving degree-zero complex. Applying this twice to two free resolutions proves the balanced calculation of the Tor functor.
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 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.
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In mathematics, particularly in the fields of algebraic topology and homological algebra, the term "double complex" refers to a structure that arises from a collection of elements arranged in a two-dimensional grid, where each entry can have additional structure, typically in the context of chain complexes. A double complex consists of a sequences of abelian groups (or modules) arranged in a grid.