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 are columns form chain 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^{[2]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 ${\displaystyle X}$ 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

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

See also[]

• Chain complex
• Derived algebraic geometry

Additional applications[]

• https://web.archive.org/web/20210708183754/http://www.dma.unifi.it/~vezzosi/papers/tou.pdf