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cohomological complex (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 $d^q$ 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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Cross-references: concept, vector spaces, topological

This is version 1 of cohomological complex, born on 2009-01-26.
Object id is 437, canonical name is CohomologicalComplex.
Accessed 283 times total.

Classification:
Physics Classification: 02. (Mathematical methods in physics)
 03. (Quantum mechanics, field theories, and special relativity )
 03.65.Fd (Algebraic methods )
Pending Errata and Addenda
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