The universal dual comodule of module in Hopf algebras. (Q1411401)

Given an algebra \(A\) over a field \(k\), the vector space \(A^0\) of all linear forms \(f\in A^*\) such that the cyclic left \(A\)-module \(Af\) is finite dimensional over \(k\) is known to be a coalgebra. The authors consider an analogous construction \(M^0\) of a right \(A^0\)-comodule for each right \(A\)-module \(M\). The main result (Theorem 4) gives the formula \((M\otimes_A N)^0\cong M^0\square_{A^0}N^0\) for modules \(M,N\) over a commutative algebra \(A\), which relates tensor and cotensor products. However, the second line of its proof reveals a conceptual misunderstanding of the nature of tensor product which sheds some doubts on the validity of this theorem.