Double complex

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

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

and the vertical differential

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

which have the compatibility relation

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

Hence a double complex is a commutative diagram of the form

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

where the rows and columns form chain complexes.

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

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

This eases the definition of Total Complexes. By setting $f_{p,q}=(-1)^{p}d_{p,q}^{v}\colon C_{p,q}\to C_{p,q-1}$ , 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.

Examples

There are many natural examples of bicomplexes that come up in nature. In particular, for a Lie groupoid, there is a bicomplex associated to it^[3]^{pg 7-8} which can be used to construct its de-Rham complex.

Another common example of bicomplexes are in Hodge theory, where on an almost complex manifold $X$ there's a bicomplex of differential forms $\Omega ^{p,q}(X)$ whose components are linear or anti-linear. For example, if $z_{1},z_{2}$ are the complex coordinates of $\mathbb {C} ^{2}$ and ${\overline {z}}_{1},{\overline {z}}_{2}$ are the complex conjugate of these coordinates, a $(1,1)$ -form is of the form