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[edit]

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