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Being motivated by open questions in gauge field theories, we consider non-standard de Rham cohomology groups for timelike compact and spacelike compact support systems. These cohomology groups are shown to be isomorphic respectively to the usual de Rham cohomology of a spacelike Cauchy surface and its counterpart with compact support. Furthermore, an analog of the usual Poincar\'e duality for de Rham cohomology is shown to hold for the case with non-standard supports as well. We apply these results to find optimal spaces of linear observables for analogs of arbitrary degree $k$ of both the vector potential and the Faraday tensor. The term optimal has to be intended in the following sense: The spaces of linear observables we consider distinguish between different configurations; in addition to that, there are no redundant observables. This last point in particular heavily relies on the analog of Poincar\'e duality for the new cohomology groups.

Optimal space of linear classical observables for Maxwell k-forms via spacelike and timelike compact de Rham cohomologies