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In mathematics, specifically Homological algebra, a double complex is a generalization of a chain complex where instead of having a Z {\displaystyle \mathbb {Z} } -grading, the objects in the bicomplex have a Z × Z {\displaystyle \mathbb {Z} \times \mathbb {Z} } -grading. The most general definition of a double complex, or a bicomplex, is given with objects in an additive category A {\displaystyle {\mathcal {A}}}. A bicomplex is a sequence of objects C p , q ∈ Ob {\displaystyle C_{p,q}\in {\text{Ob}}} with two differentials, the horizontal differential
d h : C p , q → C p + 1 , q {\displaystyle d^{h}:C_{p,q}\to C_{p+1,q}}
and the vertical differential
d v : C p , q → C p , q + 1 {\displaystyle d^{v}:C_{p,q}\to C_{p,q+1}}