Category Of Cohen-Macaulay Modules Research Articles

Relations for Grothendieck groups and representation-finiteness

For an exact category E, we study the Butler's condition “AR=Ex”: the relation of the Grothendieck group of E is generated by Auslander-Reiten conflations. Under some assumptions, we show that AR=Ex is equivalent to that E has finitely many indecomposables. This can be applied to functorially finite torsion(free) classes and contravariantly finite resolving subcategories of the module category of an artin algebra, and the category of Cohen-Macaulay modules over an order which is Gorenstein or has finite global dimension. Also we showed that under some weaker assumption, AR=Ex implies that the category of syzygies in E has finitely many indecomposables.

A categorification of Grassmannian cluster algebras

We describe a ring whose category of Cohen-Macaulay modules provides an additive categorification of the cluster algebra structure on the homogeneous coordinate ring of the Grassmannian of k-planes in n-space. More precisely, there is a cluster character defined on the category which maps the rigid indecomposable objects to the cluster variables and the maximal rigid objects to clusters. This is proved by showing that the quotient of this category by a single projective-injective object is Geiss-Leclerc-Schroer's category Sub $Q_k$, which categorifies the coordinate ring of the big cell in this Grassmannian.