Cohomologies of spaces of Schwartz test functions (Q377644)

The cohomologies of the Schwartz algebras \(\mathcal{D}\) and \(\mathcal{S}\), the test-function spaces of distribution theory, are calculated. The special cohomologies of the quotient algebras \(\mathcal{E}/\mathcal{D}\) and \(\mathcal{M}/\mathcal{S}\) (\(\mathcal{E}\) and \(\mathcal{M}\) are also test-function spaces) are found. \(\mathcal{E}/\mathcal{D}\) and \(\mathcal{M}/\mathcal{S}\) are important for calculus since the ideals \(\mathcal{D}\) and \(\mathcal{S}\) are dense in the respective algebras \(\mathcal{E}\) and \(\mathcal{M}\), and their natural quotient topologies are therefore absolutely nonseparable. Their elements (equivalence classes) do not admit canonical representatives. The author states that ``[a]ll this makes the analysis of such objects within the standard functional analysis framework very difficult and requires new ideas and methods'', and sketches why methods of classical algebra can hardly be employed here.