Tilting Modules and Exceptional Sequences for a Family of Dual Extension Algebras

Abstract

We provide a classification of generalized tilting modules and full exceptional sequences for a certain family of quasi-hereditary algebras, namely dual extension algebras of path algebras of uniformly oriented linear quivers, modulo the ideal generated by paths of length two, with their opposite algebra. An important step in the classification is to prove that all indecomposable self-orthogonal modules (with respect to extensions of positive degree) admit a filtration with standard subquotients or a filtration with costandard subquotients. Furthermore, we prove that that every generalized tilting module, not equal to the characteristic tilting modules, admits either a filtration with standard subquotients or a filtration with costandard subquotients. Since the algebras studied in this article admit a simple-preserving duality, combining these two statements reduces the problem to classifying generalized tilting modules admitting a filtration with standard subquotients. To finalize the classification of generalized tilting modules we develop a combinatorial model for the poset of indecomposable self-orthogonal modules with standard filtration with respect to the relation arising from higher extensions. This model is also used for the classification of full exceptional sequences.