Koszul duality for stratified algebras I. Balanced quasi-hereditary algebras

Abstract

We give a complete picture of the interaction between Koszul and Ringel dualities for quasi-hereditary algebras admitting linear tilting (co)resolutions of standard and costandard modules. We show that such algebras are Koszul, that the class of these algebras is closed with respect to both dualities and that on this class these two dualities commute. All arguments reduce to short computations in the bounded derived category of graded modules. 1 2. Graded quasi-hereditary algebras By N we denote the set of all positive integers. By a module we always mean a graded left module, and by grading we always mean

Ringel duality as an instance of Koszul duality

“…We believe that curve-like quasi-hereditary algebras provide an interesting class of finite-dimensional algebras. Some examples of these have already provided useful counterexamples in the work of V. Mazorchuk [Maz10] and the second author [Kül17].…”

In the bocs seat: Quasi-hereditary algebras and representation type

“…It gives an example of a basic quasi-hereditary algebra R with representation-finite filtered category but tame exact Borel subalgebra B. After constructing it, we learned that this algebra was already used as a counterexample in a different context by Mazorchuk, see [68]. The algebra is given by the following quiver: The minimal projective resolutions of standard modules are hence of the following form:…”

A generalized Koszul theory and its application

“…This question has only been studied for the case that A 0 is semisimple, see [1,2,8,19,20]. By assuming that A 0 is a self-injective algebra, and supposing that all standard modules are concentrated in degree 0 and Koszul, we get a sufficient condition for Γ to be standardly stratified with respect to op .…”

“…Therefore, A 0 e is a projective A 0 -module, and Ext * A (A 0 e, A 0 ) is an indecomposable summand of Γ. This summand corresponds to a primitive idempotent of Γ, which we still denote by e.) This question has been studied in [1,2,8,19,20]. However, in all these papers A 0 is supposed to be a semisimple algebra.…”