Ext-algebras and derived equivalences

Using derived categories, we develop an alternative approach to defining Koszulness for positively graded algebras where the degree zero part is not necessarily semisimple. The starting point for the work in this paper was to use derived categories to explain some of the results in [GRS]. In that paper the authors defined a notion of Koszulness for positively graded algebras where the degree zero part is not semisimple like it is in the classical Koszul case. When generalising a theory it is always a question which features one would like to preserve. Some basic properties of classical Koszul algebras one as a minimum would like to keep are that each Koszul algebra has a dual Koszul algebra, that the Koszul dual of the Koszul dual is isomorphic to the algebra itself, and that there is a duality between certain module categories, the objects of which are called Koszul modules. The authors of [GRS] looked at the categories of Koszul modules in the classical Koszul case, and found some additional properties they wanted to keep in the generalised setting. They used the name “T -Koszul algebras” for their generalised version of classical Koszul algebras. The T -Koszul algebras can also be viewed as a generalisation of tilting theory to the graded setting, because if one specialises to the case where the algebra is concentrated in degree zero (so basically we have an ungraded algebra), what you get is a finite-dimensional algebra together with a (Wakamatsu) cotilting module. In fact, the main purpose of [GRS] was to find a unified approach to both Koszul duality and the dualities arising from tilting theory. While the approach in [GRS] is purely module (category) theoretic, in the present paper we look at the situation from the point of view of derived categories. In the classical Koszul case the duality on the level of Koszul modules can be explained as coming from an equivalence on the level of 2000 Mathematics Subject Classification: Primary 16W50, 16S37, 18E30; Secondary 16D90.