Krull dimension of generalized Weyl algebras with noncommutative coefficients
Let \(R\) be a left Noetherian ring, \(a\) be a central element of \(R\) and \(\sigma\) be an automorphism of \(R\). The authors obtain formulae for the Krull dimension of the generalized Weyl algebra \(T=R(\sigma,a)\), this being the ring extension of \(R\) generated by two indeterminates \(x,y\) subject to the relations \(xy=\sigma(a)\), \(yx=a\) and, for all \(r\in R\), \(xr=\sigma(r)x\) and \(yr=\sigma^{-1}(r)y\). The results are too technical to present in detail here but they generalize results obtained by the first author and \textit{F. Van Oystaeyen} [J. Algebra 194, No. 2, 521-566 (1997; Zbl 0927.16002)], in the case where \(R\) is commutative, and by \textit{K. R. Goodearl} and the second author [Trans. Am. Math. Soc. 275, 833-859 (1983; Zbl 0529.16001)] for the case when \(a=1\), in which case \(T\) is a skew Laurent polynomial extension of \(R\). Methods from both papers, for example the notion of a clean module, are employed, but significant extensions of these, for example the introduction of two new kinds of cleanliness, are required. The paper has a helpful and comprehensive introduction explaining the background to the problem.
scientific article; zbMATH DE number 1563184
scientific article