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 bicomplex¹ 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 authors² 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.

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^{3pg 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 ${\displaystyle X}$ there's a bicomplex of differential forms ${\displaystyle \Omega ^{p,q}(X)}$ whose components are linear or anti-linear. For example, if ${\displaystyle z_{1},z_{2}}$ are the complex coordinates of ${\displaystyle \mathbb {C} ^{2}}$ and ${\displaystyle {\overline {z}}_{1},{\overline {z}}_{2}}$ are the complex conjugate of these coordinates, a ${\displaystyle (1,1)}$-form is of the form

${\displaystyle f_{a,b}dz_{a}\wedge d{\overline {z}}_{b}}$