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We establish a Künneth formula for some chain complexes in the categories of Fréchet and Banach spaces. We consider a complex X of Banach spaces and continuous boundary maps dn with closed ranges and prove that H (X ′) ∼= Hn(X )′, where Hn(X )′ is the dual space of the homology group of X and H(X ′) is the cohomology group of the dual complex X ′. A Künneth formula for chain complexes of nuclear Fréchet spaces and continuous boundary maps with closed ranges is also obtained. This enables us to describe explicitly the simplicial cohomology groups H(`(Z+), `(Z+)) and homology groups Hn(`(Z+), `(Z+)) of the semigroup algebra `(Z+).

A Künneth formula in topological homology and its applications to the simplicial cohomology of l1(Z+k)

A Künneth formula in topological homology and its applications to the simplicial cohomology of l¹(Z₊^k)