A Comparison of Various Types of Degenerations for Algebras

Abstract

Three different notions of geometric degeneration for finite dimensional algebras are compared. It is proved that for algebras A0 and A1 of same dimension, if A0 is a CB-degeneration of A1, then A0 is a degeneration of A1 in the classical sense; moreover, if additionally A0 and A1 are basic, then A0 is a CB-degeneration of A1 if and only if A0 is a rigid degeneration of A1. It is also shown that CB-degenerations along affine lines in the different dimension case do not differ so much from those in the equal dimension case.