REPRESENTATION THEORY AND RELATED TOPICS SEMINAR

ABSTRACT: In this talk, we will consider endomorphism rings A = End_R(M) of Cohen—Macaulay modules M over one-dimensional commutative rings R. If A has finite global dimension, then it is called a noncommutative resolution of singularities (NCR) of R (or Spec(R)).  We will give some examples of NCRs of several curve singularities, in particular for the ADE-curves, which are the plane curves of finite Cohen-Macaulay-type. For these curves one can use Auslander—Reiten theory to compute the global dimension of an endomorphism ring of a Cohen-Macaulay-module.  Moreover, we will show how the McKay correspondence for reflection groups yields a natural construction of a representation generator in this case: one can interpret ADE-curves as discriminants of finite reflection groups of GL(2,k) and then a representation generator is provided by the coordinate ring of the hyperplane (=line) arrangement of the mirrors of that reflection group. The results in this talk come from joint works with R. Buchweitz, H. Dao, B. Doherty and C. Ingalls.