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Double complex

Mathematical concept From Wikipedia, the free encyclopedia

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In mathematics, specifically Homological algebra, a double complex is a generalization of a chain complex where instead of having a -grading, the objects in the bicomplex have a -grading. 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 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

where the rows and columns form chain complexes.

Some authors[2] instead require that the squares anticommute. That is

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.

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Examples

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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 and are the complex conjugate of these coordinates, a -form is of the form

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