PlanetMath: cohomological complex of topological vector spaces

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cohomological complex of topological vector spaces

(Definition)

Definition 0.1 A cohomological complex of topological vector spaces is a pair $(E^{\bullet}, d)$ where $(E^{\bullet} = (E^q)_{q \in Z}$ is a sequence of topological vector spaces and $d = (d^q)_{q \in Z }$ is a sequence of continuous linear maps

from $E^{q}$ into $E^{q+1}$ which satisfy $d^q \circ d^{q+1} = 0$ .

Remarks

The dual complex of a cohomological complex $(E^{\bullet}, d)$ of topological vector spaces is the homological complex $(E'_{\bullet}, d')$ , where $(E'_{\bullet} = (E'_q)_{q \in Z}$ with being the strong dual of and $d' = (d'_q)_{q \in Z}$ , and also with being the transpose map of .
A cohomological complex of topological vector spaces (TVS) is a specific case of a cochain complex, which is the dual of the concept of chain complex.

"cohomological complex of topological vector spaces" is owned by bci1.

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Other names:

cohomological complex

Also defines:

dual of chain complex, cochain complex, transpose map, sequence of topological vector spaces

Keywords:

cohomological complex, cochain complex, chain complex, topological vector spaces, cohomological complex of topological vector spaces

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Cross-references: chain complex, strong, topological vector spaces, complex, linear maps, continuous, sequence
There are 6 references to this entry.

This is version 17 of cohomological complex of topological vector spaces, born on 2008-08-01, modified 2008-09-06.
Object id is 10902, canonical name is CohomologicalComplexOfTopologicalVectorSpaces.
Accessed 592 times total.

Classification:

AMS MSC:	18G35 (Category theory; homological algebra :: Homological algebra :: Chain complexes)
	13D25 (Commutative rings and algebras :: Homological methods :: Complexes)
	55N33 (Algebraic topology :: Homology and cohomology theories :: Intersection homology and cohomology)
	12G10 (Field theory and polynomials :: Homological methods :: Cohomological dimension)
	32S20 (Several complex variables and analytic spaces :: Singularities :: Global theory of singularities; cohomological properties)
	81T70 (Quantum theory :: Quantum field theory; related classical field theories :: Quantization in field theory; cohomological methods)
	55N99 (Algebraic topology :: Homology and cohomology theories :: Miscellaneous)

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