Cohomology of biflat Banach algebras with coefficients in dual bimodules (Q1921884)

A Banach bimodule \(Y\) over a Banach algebra \(A\) is said to be flat if for any admissible complex \(Q\) of Banach \(A\)-bimodules the complex \(Q\widehat\otimes Y\) is exact. A Banach algebra is said to be biflat if it is a flat bimodule. It is known that if \(X\) is a Banach \(A\)-bimodule, \(X^*\) is a dual bimodule and \(n>0\) then the cohomology groups \(H^m(A,X^*)= 0\) for \(m\geq n\) if and only if the homology groups \(H_m(A,X)=0\) for \(m\geq n\) and \(H_{m-1}(A,X)\) is a Hausdorff space. By the weak homological bidimension of a Banach algebra \(A\), the author means the least integer \(n\) such that \(H^m(A,X^*)= 0\) for all \(X\) and \(m>n\) or the symbol \(\infty\) if there is no such \(n\). The purpose of this paper is to calculate the weak homological bidimension of arbitrary biflat algebras, and to establish that \(H^m(A,X^*)= 0\) for all such algebras and \(m\geq 3\).