Quasi-hereditary algebras and generalized Koszul

We present an easily applicable sufficient condition for standard Koszul algebras to be Koszul with respect to A. If a quasi-hereditary algebra A is Koszul with respect to A, then A and the Yoneda extension algebra of A are Koszul dual in a sense explained below, implying in particular that their bounded derived categories of finitely generated graded modules are equivalent. We also prove that the extension algebra of A is Koszul in the classical sense. (C) 2013 Elsevier Inc. All rights reserved.