Algebras whose projective modules are reflexive
Let \(R\) and \(\Delta\) be rings with identity, and let \(_R M_\Delta\) be a bimodule. Let \(\text{alglat }{_RM_\Delta}= \{\alpha \in \text{End}({_RM_\Delta})\mid \alpha m \in R m\) for all \(m \in M\}\), which contains the set \(\lambda(R)\) of left multiplications by elements of \(R\). One calls \(_R M_\Delta\) reflexive if \(\text{alglat }{_RM_\Delta}= \lambda(R)\). In this paper, the authors study rings whose projective modules are reflexive, emphasizing the case where \(\Delta\) is a field, and \(R\) is a finite-dimensional \(\Delta\)-algebra. A complete classification of such algebras is not given, but it is shown that certain classes of algebras fall into this category. In previous works, the authors have shown that if a finite-dimensional \(R\) with radical \(J\) has all its projective modules reflexive, then \(R\) is split, and \(eJe = 0\) for every primitive idempotent \(e\). A class of algebras with this property is that of split hereditary algebras, for which the authors have shown previously that all projectives are reflexive. The first main result is that if \(R\) has a basic set \(e_1, \dots, e_n\) of primitive idempotents and \(e_i Je_i = 0\) for all \(i\), then every generator is reflexive. Generalizing a notion of Gordon and Small, the authors define a class of rings called quasi-piecewise domains. If a finite-dimensional algebra over an algebraically closed field is a quasi- piecewise domain, then all its projectives are reflexive. If \(R\) is a monomial algebra (zero-relation algebra), then all projectives are reflexive if and only if \(e_i Je_i = 0\) for a basic set of primitive idempotents. Finally, some further examples are found in the class of binomial algebras over an algebraically closed field of characteristic zero.
scientific article; zbMATH DE number 741482
scientific article