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
The most general definition of a double complex, or a bicomplex, is given with objects in an additive category
.
A bicomplex[1] is a sequence of objects
Ob
{\displaystyle C_{p,q}\in {\text{Ob}}({\mathcal {A}})}
with two differentials, the horizontal differential
and the vertical differential
which have the compatibility relation
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.
This eases the definition of Total Complexes.
By setting
, 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[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
there's a bicomplex of differential forms
whose components are linear or anti-linear.
For example, if
are the complex coordinates of
are the complex conjugate of these coordinates, a
-form is of the form