Modules over crossed products of a division ring by a free Abelian group. II
Let \(DA\) (\(=D*A\)) be a crossed product of a finitely generated free Abelian group \(A\) of rank \(n\) over a division ring \(D\). In this paper, the authors continue their study of finitely generated \(DA\)-modules. Their earlier work in part I of this series [J. Algebra 229, No. 1, 25-54 (2000; Zbl 0958.16028)] associated to each such module a subset \(\Delta(M)\) of the dual group \(\widehat A=\Hom(A,\mathbb{R})\cong\mathbb{R}^n\), where \(\mathbb{R}\) is the field of real numbers. With this notation, they showed that if \(\dim M\) is the largest integer \(r\leq n\) such that \(M\) contains a nonzero free \(DB\)-submodule with \(B\) a subgroup of \(A\) of rank \(r\), then \(\dim M=\text{GK}\dim M\) and \(\Delta(M)\) contains a rational convex polyhedron of dimension equal to \(\dim M\). In part II, the paper under review, the authors use this material to prove a rigidity theorem. To start with, say that two \(DA\)-modules are `similar' if they have isomorphic injective hulls and let \([M]\) denote the corresponding equivalence class of \(M\). Observe that \(\Aut(DA)\) permutes the isomorphism classes of \(DA\)-modules and hence it permutes these similarity equivalence classes. The goal here is to obtain information on \(\text{Stab}_{\Aut(DA)}[M]\), the stabilizer in \(\Aut(DA)\) of the class \([M]\). Let \({\mathcal F}_M(A)\) be the set of elements \(a\in A\) such that \(M\) has an essential \(DA\)-submodule which is \(D\langle a\rangle\)-torsion. It turns out that \({\mathcal F}_M(A)\) is a subgroup of \(A\) with factor group \(A/{\mathcal F}_M(A)\) torsion-free. Note that any automorphism of \(DA\) determines an automorphism of \(D\) and also one of \(A\). Moreover, if this automorphism stabilizes \([M]\), then its action on \(A\) leaves \({\mathcal F}_M(A)\) invariant and hence it yields an automorphism of \(A/{\mathcal F}_M(A)\). The main result of this paper is Theorem B. Let \(M\) be a finitely generated \(DA\)-module and assume that \(M\) contains no nonzero submodule induced from \(DB\) for any subgroup \(B\) of infinite index in \(A\). Then \(\text{Stab}_{\Aut(DA)}[M]\) has finite image in \(\Aut(A/{\mathcal F}_M(A))\). -- In a forthcoming paper, the authors use this result to obtain information on automorphisms of certain infinite solvable groups.
scientific article; zbMATH DE number 1836733