The Jordan-Zassenhaus theorem and direct decompositions
Recall, that the Jordan-Zassenhaus theorem states that, if \(R\) is a Dedekind domain whose field \(Q\) of quotients is a global field, then for each \(R\)-order \(S\) in a semisimple algebra over \(Q\), and for each positive integer \(n\), there are only finitely many isomorphism classes of left \(S\)-lattices of rank \(\leq n\). For a Dedekind domain \(R\) of characteristic 0 the Jordan-Zassenhaus theorem holds if and only if for every non-zero ideal \(I\) of \(R\), the factor-ring \(R/I\) is finite, and every finite rank torsion-free \(R\)-module has but a finite number of non-isomorphic direct summands (Theorem 5.2). For a Dedekind \(R\) of characteristic 0 the following are equivalent: (i) every torsion-free \(R\)-module of finite rank has, up to equivalence, but a finite number of direct decompositions; (ii) for every \(R\)-order \(S\), the class \(\text{Cl}(S)\) is finite, and torsion-free \(R\)-modules of finite rank have the almost cancellation property; (iii) for every \(R\)-order \(S\), the genus classes of \(S\)-lattices are finite; (iv) every \(R\)-order \(S\) satisfies: (a) for every ideal \(I\) of \(S\) with \(I\cap R\neq 0\), the subgroup \(\overline{U(S)}\) has finite index in \(U(S/I)\) and (b) there is \(0\neq r\in R\) such that every \(S\)-lattice in the genus of \(S\) is \(S\)-isomorphic to an \(S\)-submodule between \(rS\) and \(S\); (v) every \(R\)-order \(S\) satisfies: (a) for every ideal \(I\) of \(S\) with \(I\cap R\neq 0\), the subgroup \(\overline{U(S)}\) has finite index in \(U(S/I)\) and (b) the genus of \(S\) is finite (Theorems 5.1 and 4.3).
scientific article; zbMATH DE number 1512989
scientific article