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cohomological complex of topological vector spaces
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(Definition)
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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 $d^q$ 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 $E'_q$ being the strong dual of $E^q$ and $d' = (d'_q)_{q \in Z}$ , and also with $d'_q $ being the transpose map of $d^q$ .
- 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.
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"cohomological complex of topological vector spaces" is owned by bci1.
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See Also: homological complex of topological vector spaces, chain complex, categorical sequence, tangential Cauchy-Riemann complex of ![$C^{\infty)$]( "$C^{\infty)$") -smooth forms, tangential Cauchy-Riemann complex of smooth forms
| Other names: |
cohomological complex |
| Also defines: |
dual of chain complex, cochain complex, transpose map, sequence of topological vector spaces, sequence of continuous linear maps |
| 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
There are 8 references to this entry.
This is version 19 of cohomological complex of topological vector spaces, born on 2008-08-01, modified 2009-06-01.
Object id is 10902, canonical name is CohomologicalComplexOfTopologicalVectorSpaces.
Accessed 2365 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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Pending Errata and Addenda
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