Module Varieties over Canonical Algebras

Abstract

The main purpose of this paper is the study of module varieties over the class of canonical algebras, providing a rich source of examples of varieties with interesting properties. Our main tool is a stratification of module varieties, which was recently introduced by Richmond. This stratification does not require a precise knowledge of the module category. If it is finite, then it provides a method to classify irreducible components. We determine the canonical algebras for which this stratification is finite. In this case, we describe the algorithm for calculating the dimension of the variety and the number of irreducible components of maximal dimension. For an infinite family of examples we give easy combinatorial criteria for irreducibility, Cohen-Macaulay and normality.  2001 Elsevier Science

On the additive categories of generalized standard almost cyclic coherent Auslander–Reiten components

“…In particular, the geometry of module varieties with separating families of tubes, or more generally coils, has attracted much attention (see [4][5][6][7]26,27] for some results in this direction).…”

Closures in varieties of representations and irreducible components

“…(On the side, we mention that Rep S (3) is not contained in C 2 because the sequences S (2) and S (3) are not comparable under the dominance order.) The sequence S (4) fails to be realizable for s = 1.…”

Understanding Finite Dimensional Representations Generically

“…Bounds on the number of components for certain tame algebras Λ may be obtained from an interesting stratification of the varieties Rep d (Λ) due to Richmond [31]. Barot and Schröer further explored these stratifications over canonical algebras in [3]. where L is a positive integer.…”