Continuous Frobenius Categories

We introduce continuous Frobenius categories. These are topological categories which are constructed using representations of the circle over a discrete valuation ring. We show that they are Krull-Schmidt with one indecomposable object for each pair of (not necessarily distinct) points on the circle. By putting restrictions on these points we obtain various Frobenius subcategories. The main purpose of these Frobenius categories is to give a precise and elementary description of the triangulated structure of their stable categories, which are cluster categories for certain values of the parameters. These continuous cluster categories are being developed in concurrently written papers. The standard construction of a cluster category of a hereditary algebras is to take the orbit category of the derived category of bounded complexes of finitely generated modules over the algebra: CH ∼= D(modH)/F where F is a triangulated autoequivalence of Db(modH) [2]. In this paper we construct continuous versions of the cluster categories of type An. These continuous cluster categories are “continuously triangulated” categories having uncountably many indecomposable objects and containing the finite and countable cluster categories of type An and A∞ as subquotients. Cluster categories of type An and A∞ were also studied in [3], [6], [12]. The reason for the term continuous in the names of the categories is the fact that the categories that we define and consider in this paper are topological categories with continuous structure maps (Section 0). The continuity requirement implies that there are two possible topologically inequivalent triangulations of the continuous cluster category given by the two 2-fold covering spaces of the Moebius band: connected and disconnected. We consider both cases (Remarks 3.1.8, 3.4.2). The term cluster in the names of the categories is justified in [8] where it is show that the category Cπ has a cluster structure where cluster mutation is given using the triangulated structure (see [1]) and that the categories Cc also have a cluster structure for specific values of c. For the categories Cφ, we have partial results (Cφ has an m-cluster structure in certain cases). This paper is the first in a series of papers. The main purpose of this paper is to give a concrete and self-contained description of the triangulated structures of these continuous cluster categories being developed in concurrently written papers [8, 9]. We will use representations of the circle over a discrete valuation ring R to construct continuous Frobenius R-categories Fπ, Fc and Fφ whose stable categories (triangulated by [4]) are isomorphic to the continuous categories Cπ, Cc and Cφ, thus inducing continuous triangulated structure on these topological K-categories (K = R/m). In Section 1. we define representations of the circle; a representation of the circle S1 = R/2πZ over R is defined to be collection of R-modules V [x] at every point x ∈ S1 and Date: July 4, 2012. 2000 Mathematics Subject Classification. 18E30:16G20. The first author is supported by NSA Grant 111015.