Equivalences of derived categories for selfinjective algebras

Equivalences of Derived Categories for Symmetric Algebras

It is about a decade since Broué made his celebrated conjecture [2] on equivalences of derived categories in block theory: that the module categories of a block algebra A of a finite group algebra and its Brauer correspondent B should have equivalent derived categories if their defect group is abelian. Since then, character-theoretic evidence for the conjecture has accumulated rapidly, but unti...

Cluster Categories and Selfinjective Algebras: Type A

We show that the stable module categories of certain selfinjective algebras of finite representation type having tree class A n are actually u-cluster categories. Since their introduction in [6], [7], cluster categories have become a central topic in representation theory. They provide the framework for the representation-theoretic approach to the highly successful theory of cluster algebras, a...

Derived Categories , Derived Equivalences and Representation Theory

Deenition: A derived category ... is when you take complexes seriously! (L.L. Scott Sc]) The aim of this chapter is to give a fairly elementary introduction to the (not very elementary) subject of derived categories and equivalences. Especially, we emphasize the applications of derived equivalences in representation theory of groups and algebras in order to illustrate the importance and usefuln...

Derived equivalences and Gorenstein algebras

In this note, we introduce the notion of Gorenstein algebras. Let R be a commutative Gorenstein ring and A a noetherian R-algebra. We call A a Gorenstein R-algebra if A has Gorenstein dimension zero as an R-module (see [2]), add(D(AA)) = PA, where D = HomR(−, R), and Ap is projective as an Rpmodule for all p ∈ Spec R with dim Rp < dim R. Note that if dim R = ∞ then a Gorenstein R-algebra A is p...

The Derived Equivalence Classification of Representation-finite Selfinjective Standard Algebras