Double complex

In mathematics, specifically Homological algebra, a double complex is a generalization of a chain complex where instead of having a ${\displaystyle \mathbb {Z} }$-grading, the objects in the bicomplex have a ${\displaystyle \mathbb {Z} \times \mathbb {Z} }$-grading. The most general definition of a double complex, or a bicomplex, is given with objects in an additive category ${\displaystyle {\mathcal {A}}}$. A bicomplex[1] is a sequence of objects ${\displaystyle C_{p,q}\in {\text{Ob}}({\mathcal {A}})}$ with two differentials, the horizontal differential

${\displaystyle d^{h}:C_{p,q}\to C_{p+1,q}}$

and the vertical differential

${\displaystyle d^{v}:C_{p,q}\to C_{p,q+1}}$

which have the compatibility relation

${\displaystyle d_{h}\circ d_{v}=d_{v}\circ d_{h}}$

Hence a double complex is a commutative diagram of the form

${\displaystyle {\begin{matrix}&&\vdots &&\vdots &&\\&&\uparrow &&\uparrow &&\\\cdots &\to &C_{p,q+1}&\to &C_{p+1,q+1}&\to &\cdots \\&&\uparrow &&\uparrow &&\\\cdots &\to &C_{p,q}&\to &C_{p+1,q}&\to &\cdots \\&&\uparrow &&\uparrow &&\\&&\vdots &&\vdots &&\\\end{matrix}}}$

where the rows and columns form chain complexes.

Some authors[2] instead require that the squares anticommute. That is

${\displaystyle d_{h}\circ d_{v}+d_{v}\circ d_{h}=0.}$

This eases the definition of Total Complexes. By setting ${\displaystyle f_{p,q}=(-1)^{p}d_{p,q}^{v}\colon C_{p,q}\to C_{p,q-1}}$, we can switch between having commutativity and anticommutativity. If the commutative definition is used, this alternating sign will have to show up in the definition of Total Complexes.