The cohomology group of weak entwining structure. (Q995724)

Weak entwining structures have been introduced by the reviewer and De Groot, and generalize classical entwining structures. In this note, Hochschild cohomology associated to a weak entwining structure \((A,C,\psi)\) with values in an \(A\)-bimodule \(M\) is introduced. It is well-known that a certain direct factor \(\overline{A\otimes C}\) of \(A\otimes C\) is an \(A\)-coring. The main result states that \(\overline{A\otimes C}\) is projective as an \(A\)-bimodule if and only if the first Hochschild cohomology group \(H^1_\psi(A,M)=0\) for every \(A\)-bimodule \(M\). The authors then consider the special case where the weak entwining structure is associated to a weak coalgebra Galois extension.