Adam Hajduk

A Comparison of Various Types of Degenerations for Algebras

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.

On Generalized CB-Degenerations of Algebras

We introduce a concept generalizing classical degenerations of algebras (defined by structure constants) and Crawley-Boevey degenerations introduced in [3]. We prove that if A 0 is such a generalized degeneration of A 1 and the algebras have equal dimensions, then A 0 is a degeneration of A 1 in the classical sense.

Partial Orders and CB-Degeneration Relation

We show that the relation ≤cb, induced from the CB-degeneration relation on the disjoint union ∐d∈ℕ(algd(k)/≅), is a partial order, where for d ∈ ℕ the set algd(k)/≅ consists of the isomorphism classes of k-algebra structures on the space kd.