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.
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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.