Smash products and comodules of linear maps (Q1174668)

Let \(H\) be a finite Hopf algebra over the commutative ring \(k\), \(A\) a right \(H\)-comodule algebra, \(U=H\otimes A\) and \(\hbox{End}^ H_ A(U)\) the set of \(A\)-linear and \(H\)-colinear maps from \(U\) to \(U\). It is shown that \(\hbox{End}^ H_ A(U)\#H\simeq\hbox{End}_ A(U)\) and \(A\#H^*\simeq\hbox{End}^ H_ A(U)\). These are generalizations of similar results obtained by Năstăsescu and Rodinò in the case of finite group-gradings.