Homological smoothness and deformations of generalized Weyl algebras

TL;DR

This paper establishes that a generalized Weyl algebra's homological smoothness is equivalent to its defining polynomial having no multiple roots, and it explores formal deformations in characteristic zero.

Contribution

It proves the converse of Bavula's result linking smoothness to polynomial roots and studies formal deformations of these algebras.

Findings

Homological smoothness of GWA is characterized by simple roots of the polynomial.

The converse of Bavula's smoothness criterion is proven.

Formal deformations are analyzed in characteristic zero.

Abstract

It is an immediate conclusion from Bavula's papers \cite{Bavula:GWA-def}, \cite{Bavula:GWA-tensor-product} that if a generalized Weyl algebra $A=\kk[z;\lambda,\eta,\varphi(z)]$ is homologically smooth, then the polynomial $\varphi(z)$ has no multiple roots. We prove in this paper that the converse is also true. Moreover, formal deformations of $A$ are studied when $\kk$ is of characteristic zero.