The K\"unneth formula for nuclear $DF$-spaces and Hochschild cohomology

TL;DR

This paper proves a duality relation between homology and cohomology in nuclear Fréchet spaces, and applies it to establish Künneth formulas for Hochschild and cyclic cohomology of nuclear $DF$-spaces and $ ext{Fréchet}$-algebras.

Contribution

It establishes a topological isomorphism between duals of homology groups and cohomology groups for complexes of nuclear Fréchet spaces, and derives Künneth formulas for Hochschild and cyclic cohomology in this setting.

Findings

Proved duality between homology and cohomology in nuclear Fréchet spaces.

Established Künneth formulas for Hochschild and cyclic cohomology of nuclear $DF$-spaces.

Explicitly described cohomology groups of tensor products of $ ext{Fréchet}$-algebras.

Abstract

We consider complexes $(\X, d)$ of nuclear Fr\'echet 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 K\"{u}nneth formula for the cohomology of complete nuclear $DF$-complexes and in the K\"{u}nneth formula for continuous Hochschild cohomology of nuclear $\hat{\otimes}$-algebras which are Fr\'echet 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 Fr\'echet spaces or nuclear $DF$-spaces.