The Künneth formula for nuclear $DF$-spaces and Hochschild cohomology

Abstract

We consider complexes $(\X, d)$ of nuclear Fréchet spaces and continuous boundary maps $d_n$ with closed ranges and prove that, up to topological isomorphism, $ (H_{n}(\X, d))^*$ $\iso$ $H^{n}(\X^*,d^*),$ where $(H_{n}(\X,d))^*$ is the strong dual space of the homology group of $(\X,d)$ and $ H^{n}(\X^*,d^*)$ is the cohomology group of the strong dual complex $(\X^*,d^*)$. We use this result to establish the existence of topological isomorphisms in the Künneth formula for the cohomology of complete nuclear $DF$-complexes and in the Künneth formula for continuous Hochschild cohomology of nuclear $\hat{\otimes }$-algebras which are Fréchet spaces or $DF$-spaces for which all boundary maps of the standard homology complexes have closed ranges. We describe explicitly continuous Hochschild and cyclic cohomology groups of certain tensor products of $\hat{\otimes }$-algebras which are Fréchet spaces or nuclear $DF$-spaces.

31 pages