On the homological algebra of clusters, quivers, and triangulations

Abstract

This thesis is comprised of three parts. Chapter one contains background material detailing some important aspects of category theory and homological algebra. Beginning with abelian categories, we introduce triangulated categories, the homotopy category and give the construction of the derived category. Di erential graded algebras, basic Auslander- Reiten theory, and the Cluster Category of Dynkin Type An are also introduced, which all play a major role in chapters two and three. In [21], the cluster category D of type A1, with Auslander-Reiten quiver ZA1, is introduced. Slices in the Auslander-Reiten quiver of D give rise to direct systems; the homotopy colimit of such direct systems can be computed and these \Pr ufer objects" can be adjoined to form a larger category. It is this larger category, D; which is the main object of study in chapter two. We show that D inherits a nice geometrical structure from D; \arcs" between non-neighbouring integers on the number line correspond to indecomposable objects, and in the case of D we also have arcs to in nity which correspond to the Pr ufer objects. During the course of chapter two, we show that D is triangulated, compute homs, investigate the geometric model, and we conclude by computing the cluster tilting subcategories of D. Frieze patterns of integers were studied by Conway and Coxeter, see [13] and [14]. Let C be the cluster category of Dynkin type An. Indecomposables in C correspond to diagonals in an (n + 3)-gon. Work done by Caldero and Chapoton showed that the Caldero-Chapoton map (which is a map dependent on a xed object R of a category, and which goes from the set of objects of that category to Z), when applied to the objects of C can recover these friezes, see [10]. This happens precisely when R corresponds to a triangulation of the (n + 3)-gon, i.e. when R is basic and cluster tilting. Later work (see [6], [22]) generalised this connection with friezes further, now to d-angulations of the (n + 3)-gon with R basic and rigid. In chapter three, we extend these generalisations further still, to the case where the object R corresponds to a general Ptolemy diagram, i.e. R is basic and add(R) is the most general possible torsion class (where the previous e orts have focused on special cases of torsion classes).