Generalized Weyl Algebras Are Tensor Krull Minimal

Abstract

Formulae for calculating the Krull dimension of noetherian rings obtained by the authors and their collaborators are used to calculate Krull dimension for certain classes of algebras. An F-algebra T is said to be tensor Krull minimal (TKM) with respect to a class of F-algebras Ω if K(T⊗B)=K(T)+K(B), for each B∈Ω. We show that generalized Weyl algebras over affine commutative F-algebras, where F is an uncountable algebraically closed field, are TKM with respect to the class of countably generated left noetherian F-algebras. This simplifies the task of calculating many Krull dimensions. In addition, we develop an improved formula for the Krull dimension of a skew Laurent extension D[x,x−1;σ], where D is a polynomial algebra over an algebraically closed field, and σ is an affine automorphism. Finally, we calculate the Krull dimension of the noetherian down–up algebras introduced by Benkart.