Reflexive modules are not closed under submodules (Q2782427)

The object of this paper is to prove that the classes of reflexive modules with respect to a cotilting bimodule fail to be closed under submodules. The author considers an infinite-dimensional generalized Kronecker algebra \(A\) over an algebraically closed field \(K\) and shows that \(_AA_A\) is a cotilting bimodule which has very few reflexive modules, as these are just the finitely generated projective modules. As a consequence of this fact, the only reflexive modules with respect to \(_AA_A\) with the property that all the submodules are also reflexive, are those that are finite-dimensional over \(K\). From this it follows that, for such an algebra, both classes of reflexive modules are not closed under submodules. As an application, the author also exhibits some surprising properties of this construction. For example, it is shown that, for any nonzero cardinal \(c\) which is less than or equal to the dimension of \(A\) over \(K\), there exists an indecomposable cyclic \(A\)-module \(M\) of dimension \(c\) over \(K\), such that \(\Hom_A(M,A)=0\), and \(\text{Ext}^1_A(M,A)\) is a free module of uncountable rank.NEWLINENEWLINEFor the entire collection see [Zbl 0974.00038].