cohomological complex | Physics Library

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 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.

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