Julia Sauter (Bielefeld), Shifting algebras of positive dominant dimension and desingularizations of orbit closures.

Abstract:

To any finite dimensional algebra of positive dominant dimension we associate tilting modules by taking cosyzygies of the algebra and add a maximal projective-injective summand of the algebra. These tilting modules will be referred to as "shifted" modules and their endomorphism algebras as shifted algebras. These algebras have some interesting properties: Firstly they allow us to give a tilted version of the Morita-Tachikawa correpsondence where you have to replace endomorphism rings of generator-cogenerators by endomorphism rings of certain complexes in the homotopy category. Secondly they are the natural generalization of the projective-quotient algebras associated to algebras of finite-representation type (in previous work by Crawley-Boevey-Sauter) and therefore can be used in geometric applications as desingularizations of orbit closures. The main advantage of the shifted algebras over the endomorphism rings of generator-cogenerators is that they come with a recollement realizing the dual of the shifted module as the intermediate extension of the generator-cogenerator. In the desingularization of orbit closures (of loc. cit.) we can replace "finite-representation type of the algebra" by the much weaker property that the given module is quotient-finite (meaning that there are only finitely many isomorphism classes of quotient modules). This is joint work in progress with Matt Pressland.