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We consider complexes $(\X, d)$ of nuclear Fr\'echet spaces and continuousboundary maps $d_n$ with closed ranges and prove that, up to topologicalisomorphism, $ (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 topologicalisomorphisms in the K\"{u}nneth formula for the cohomology of complete nuclear$DF$-complexes and in the K\"{u}nneth formula for continuous Hochschildcohomology of nuclear $\hat{\otimes}$-algebras which are Fr\'echet spaces or$DF$-spaces for which all boundary maps of the standard homology complexes haveclosed ranges. We describe explicitly continuous Hochschild and cycliccohomology groups of certain tensor products of $\hat{\otimes}$-algebras whichare Fr\'echet spaces or nuclear $DF$-spaces.

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