Degenerations and other partial orders on the space of representations of algebras

Abstract

Let K be a field and¤be an artin K-algebra. Let r epd¤represent the set of all¤-modules with the length equal to a natural number d as a K-vector space. The set of modules r epd¤ is equipped with the action of the general linear group. The corresponding Zariski-topology for algebraically closed field K then induce a partial order on r epd¤, which is called degeneration order and it is denoted by ·deg . Here for M and N, ¤- modules, the notion M ·deg N mean that the orbit of N under the action of general linear group is contained in the closure of the orbit of M under the same group action. Another partial order on r epd¤ first showed by Riedtmann, is the virtual degeneration order, which is denoted by ·vdeg , are given by M ·vdeg N, if there is a ¤-module X such that M © X ·deg N © X. There are known examples where these two partial orders do not coincide. If K is an algebraically closed field, there is a geometric interpretation of these notions. However, there is also a module theoratical interpretation, which can be generalized to the general settings with K a commutative artin ring. Let ¡ be the Kronecker quiver 1â2 and ¤Æ Z2¡ be the path algebra of ¡ over the field Z2 with two elements. In this work all degenerations between isomrphism classes of modules over ¤ of dimension vector (1, 1), (2,2) and (3,3) are determined and the Hasse diagrams of the corresponding partial orders are given.