Gelfand-Kirillov dimension in some crossed products
Let \(k\) be a field. The author considers a PBW-extension \(A=R[\theta_1,\dots,\theta_n]\) of an algebra \(R\) over \(k\). This is a ring extension \(A\supset R\) generated by elements \(\{\theta_1,\dots,\theta_n\}\) such that (i) \([\theta_i,r]\in R\) for each \(i=1,\dots,n\); (ii) \([\theta_i,\theta_j]\in R+R\theta_1+\cdots+R\theta_n\); (iii) \(A\) is free as (left and right) \(R\)-module with basis \(\theta_1^{i_1}\cdots\theta_n^{i_n}\), where \(i_j\) are positive integers. Let \(\delta_i\) be the respective derivation of \(R\), i.e. \(\delta_i(r)=[\theta_i,r]\) for all \(r\in R\). Theorem. Let \(R\) be locally finite-dimensional over \(\Delta=\{\delta_1,\dots,\delta_n\}\). Then the Gelfand-Kirillov dimensions are connected as \(\text{GKdim}(A)=\text{GKdim}(R)+n\). Earlier, the same fact was known for a finitely generated ring \(R\) over \(k\), now this condition is replaced by the local finite dimensionality over \(\Delta\). From this the author deduces some results concerning incomparability, prime length and Tauvel's height formula for prime ideals in the crossed products \(R*G\), where \(G\) is a finite-dimensional Lie algebra acting by derivations on \(R\). Similar results are obtained for \(R*G\), where \(G\) is a free Abelian group of finite rank.
scientific article; zbMATH DE number 1865996