Geometry of Directing Modules over Tame Algebras

Abstract

Throughout the paper K denotes a fixed algebraically closed field. By an Ž algebra we mean a finite dimensional K-algebra associative, with an . identity and by a module a finite dimensional left A-module. The class of algebras may be divided into two disjoint subclasses. One class consists of tame algebras for which the indecomposable modules occur, in each dimension, in a finite number of discrete and a finite number of one-parameter families. The second class is formed by the wild algebras whose representation theory is as complicated as the study of finite dimensional vector spaces together with two noncommuting endomorphisms, for which the classification up to isomorphism is a well-known unsolved problem. Hence, we can realistically hope to describe modules only for tame algebras. Given an algebra A and a nonnegative vector d in Ž . the Grothendieck group K A of A, it is an interesting task to study the 0 Ž . affine variety mod d of A-modules of dimension-vector d and the action A Ž . of the corresponding product G d of general linear groups. For example, Ž . we may ask when the variety mod d is irreducible, smooth, complete A intersection, Gorenstein, Cohen]Macaulay, normal, . . . . The main aim of this paper is to describe the geometry of module Ž . varieties mod d for the dimension-vectors d of arbitrary directing modA w x ules over tame algebras. Recall that following 20 an indecomposable A-module M is said to be directing if it does not belong to a cycle M a M a ??? a M a M of nonzero nonisomorphisms between inde1 r composable A-modules. Directing modules have played an important role