On projective modules over quantum polynomials
This is partly an expository article on various skew polynomial rings and related rings but it also includes new results on a particular ring denoted \(D_{Q,\alpha}[X_1,\ldots,X_n,X_{i_1}^{-1},\ldots,X_{i_r}^{-1}]\). This is constructed from a division algebra \(D\) over a field \(k\) using \(k\)-automorphisms \(\alpha_i\) of \(D\), \(1\leq i\leq n\), and non-zero elements \(q_{ij}\in D\), \(1\leq i,j\leq n\), satisfying the equations \(q_{ii}=q_{ij}q_{ji}=Q_{ijr}Q_{jri}Q_{rij}=1\), where \(Q_{ijr}=q_{ij}\alpha_j(q_{ir})\). The \(n\) indeterminates \(X_i\) are adjoined to \(D\) subject to the relations \(X_iz=\alpha_i(z)X_i\), for \(z\in D\), and \(X_iX_j=q_{ij}X_jX_i\) and then \(r\) of these indeterminates are inverted. In earlier papers of the author [Mat. Sb. 185, No. 7, 3-12 (1994; Zbl 0849.16003) and Mat. Zametki 59, No. 4, 497-503 (1996; Zbl 0879.16013)], a certain freeness condition was imposed on the elements \(q_{ij}\). The new results in the present paper, on the freeness of projective modules of sufficiently large rank, on Krull dimension, and on simplicity when \(r=n\) are similar to results from these earlier papers but are proved under different conditions on the \(q_{ij}\)'s.
scientific article; zbMATH DE number 1295561
scientific article