mathematics Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes. These topics are represented in modern mathematics ...

, specifically

Homological algebra Homological algebra is the branch of mathematics that studies homology (mathematics), homology in a general algebraic setting. It is a relatively young discipline, whose origins can be traced to investigations in combinatorial topology (a precurs ...

, a double complex is a generalization of a

chain complex In mathematics, a chain complex is an algebraic structure that consists of a sequence of abelian groups (or module (mathematics), modules) and a sequence of group homomorphism, homomorphisms between consecutive groups such that the image (mathemati ...

where instead of having a

\mathbb

-grading, the objects in the bicomplex have a

\mathbb\times\mathbb

-grading. The most general definition of a double complex, or a bicomplex, is given with objects in an

additive category In mathematics, specifically in category theory, an additive category is a preadditive category C admitting all finitary biproducts. Definition A category C is preadditive if all its hom-sets are abelian groups and composition of m ...

\mathcal

. A bicomplex is a sequence of objects

C_ \in \text(\mathcal)

with two differentials, the horizontal differential

$d^h: C_ \to C_$

and the vertical differential

$d^v:C_ \to C_$

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
& & \vdots & & \vdots & & \\
& & \uparrow & & \uparrow & & \\
\cdots & \to & C_ & \to & C_ & \to & \cdots \\
& & \uparrow & & \uparrow & & \\
\cdots & \to & C_ & \to & C_ & \to & \cdots \\
& & \uparrow & & \uparrow & & \\
& & \vdots & & \vdots & & \\

\end$

where the rows and columns form chain complexes. Some authors 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_ = (-1)^p d^v_ \colon C_ \to C_

, 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 In mathematics, a Lie groupoid is a groupoid where the set \operatorname of objects and the set \operatorname of morphisms are both manifolds, all the category operations (source and target, composition, identity-assigning map and inversion) are smo ...

, there is a bicomplex associated to it^{pg 7-8} which can be used to construct its de-Rham complex. Another common example of bicomplexes are in

Hodge theory In mathematics, Hodge theory, named after W. V. D. Hodge, is a method for studying the cohomology groups of a smooth manifold ''M'' using partial differential equations. The key observation is that, given a Riemannian metric on ''M'', every cohom ...

, where on an

almost complex manifold In mathematics, an almost complex manifold is a smooth manifold equipped with a smooth linear complex structure on each tangent space. Every complex manifold is an almost complex manifold, but there are almost complex manifolds that are not compl ...

X

there's a bicomplex of differential forms

\Omega^(X)

whose components are linear or anti-linear. For example, if

z_1,z_2

are the complex coordinates of

\mathbb^2

and

\overline_1,\overline_2

are the complex conjugate of these coordinates, a

(1,1)

-form is of the form

$f_dz_a\wedge d\overline_b$

Examples

See also

Additional applications