Double complex (original) (raw)
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 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.
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| dbo:abstract | 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 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 instead require that the squares anticommute. That is This eases the definition of . 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. (en) |
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| rdfs:comment | 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 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. (en) |
| rdfs:label | Double complex (en) |
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