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Crossed products and finitely presented groups. (English) Zbl 0978.16028
Let \(F\) be a field, let \(n\) be a positive integer and let \({\mathbf q}=(q_{ij})\) be an \(n\times n\) matrix with entries from \(F\setminus\{0\}\) such that \(q_{ij}=q^{-1}_{ji}\) and \(q_{ii}=1\) for all \(i,j\). Localised quantum \(n\)-space \(A({\mathbf q},F)\) is the \(F\)-algebra generated by \(X^{\pm 1}_1,\ldots,X^{\pm 1}_n\) subject to the relations \(X_iX_j=q_{ij}X_jX_i\) for all \(i,j\). These algebras are fundamental objects in the study of quantum groups; they also arise frequently in work on infinite groups and their group algebras, since \(A({\mathbf q},F)\) may be regarded as the crossed product \(F*G\) of a free Abelian group \(G\) with \(F\) central. Define \(t\) to be the maximum of the ranks of the subgroups \(H\) of \(G\) for which \(F*H=FH\), the ordinary group algebra. J. C. McConnell and J. J. Pettit [J. Lond. Math. Soc., II. Ser. 38, 47-55 (1988; Zbl 0652.16007)] observed that \(t\) is a lower bound for the global and Krull dimensions of \(A({\mathbf q},F)\), and conjectured that equality holds in both cases. The main theorem of this paper confirms both these equalities, a beautiful result which improves on earlier special cases obtained by a number of authors. The paper also includes applications to finitely presented groups which are somewhat too technical to state here.

MSC:
16S35 Twisted and skew group rings, crossed products
20F05 Generators, relations, and presentations of groups
16E10 Homological dimension in associative algebras
16W35 Ring-theoretic aspects of quantum groups (MSC2000)
16P60 Chain conditions on annihilators and summands: Goldie-type conditions
20F16 Solvable groups, supersolvable groups
16S34 Group rings
20C07 Group rings of infinite groups and their modules (group-theoretic aspects)
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