mathematics Mathematics is a field of study that discovers and organizes methods, Mathematical theory, theories and theorems that are developed and Mathematical proof, proved for the needs of empirical sciences and mathematics itself. There are many ar ...

, 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 modules) and a sequence of homomorphisms between consecutive groups such that the image of each homomorphism is contained in the kernel o ...

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 There are two equivalent definitions of an additive category: One as a category equipped wit ...

\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 Total may refer to: Mathematics * Total, the summation of a set of numbers * Total order, a partial order without incomparable pairs * Total relation, which may also mean ** connected relation (a binary relation in which any two elements are co ...

. 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 sm ...

, 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 coho ...

, 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 comple ...

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