Modules over crossed products of a division ring by a free Abelian group. I
Let \(DA\) be a crossed product of \(A\), a free Abelian group of finite rank \(n\), over the division ring \(D\). This paper studies \(DA\)-modules \(M\) and uses the results to obtain interesting group theoretic applications. The main results are quite technical, so we will just state one. To start with, the dimension of \(M\), \(\dim M\), is defined to be the largest integer \(r\) such that \(M\) contains a nonzero free \(DB\)-submodule for some subgroup \(B\) of \(A\) with the rank of \(B\) equal to \(r\). This turns out to be the same as the Gelfand-Kirillov dimension of \(M\). Next, let \(R\) denote the additive group of real numbers and let \(\widehat A=\Hom(A,R)=R^n\) denote the set of characters \(\chi\colon A\to R\). If \(\chi\in\widehat A\), we say that \(M\) has a nontrivial \(\chi\)-filtration if there exists a family of \(D\)-spaces \(\{M_\mu\mid\mu\in R\}\) satisfying (1) \(M_\mu\supseteq M_\lambda\) if and only if \(\lambda\geq\mu\), (2) \(M=\bigcup_\mu M_\mu\) and \(M\neq M_\mu\), and (3) \(M_\mu\overline a=M_{\mu+\chi(a)}\) for all \(a\in A\). Then \(\Delta(M)\) is defined to be the subset of \(\widehat A\) consisting of \(0\) and those characters \(\chi\) such that \(M\) has a nontrivial \(\chi\)-filtration. With this notation, Theorem 4.4 asserts that \(\Delta(M)\subseteq\widehat A=R^n\) contains a rational convex polyhedron of dimension \(r=\dim M\). Note that a convex polyhedron is a finite intersection of closed half-spaces, and it is rational if all the defining half spaces have boundaries that are rational subspaces.
scientific article; zbMATH DE number 1496002
scientific article