 
Summary: TOPSTABLE DEGENERATIONS OF FINITE
DIMENSIONAL REPRESENTATIONS I
Birge HuisgenZimmermann
Dedicated to Claus Michael Ringel on the occasion of his sixtieth birthday
Abstract. Given a finite dimensional representation M of a finite dimensional algebra, two
hierarchies of degenerations of M are analyzed in the context of their natural orders: the poset
of those degenerations of M which share the top M/JM with M here J denotes the radical of
the algebra and the subposet of those which share the full radical layering
`
JlM/Jl+1M
´
l0
with M. In particular, the article addresses existence of proper topstable or layerstable
degenerations more generally, it addresses the sizes of the corresponding posets including
bounds on the lengths of saturated chains as well as structure and classification.
1. Introduction
Let be a basic finite dimensional algebra over an algebraically closed field K. Our ob
jective is to understand major portions of the degeneration theory of the finite dimensional
representations of . This line of inquiry was triggered by Gabriel's and Kac's initial ex
ploitation of the affine variety Mod
