The genus of a module. II: Roiter's theorem, power cancellation and extension of scalars
In this interesting paper the author extends classical results on left \(\Lambda\)-lattices, where \(\Lambda\) is an order over the Dedekind domain \(R\) in a separable algebra over the quotient field \(K\) of \(R\), to the case of finitely generated left \(\Lambda\)-modules. More precisely, there exists an integer \(n=n(\Lambda)\), such that for any two finitely generated left \(\Lambda\)-modules \(M\) and \(N\), which are locally isomorphic, one has \(M^{(n)}\cong N^{(n)}\). If \(R\) satisfies the Jordan-Zassenhaus theorem, then \(n\) depends only on the skewfields in \(K\Lambda\) and the \(R\)-conductor from a maximal order \(\Gamma\supset \Lambda\). In the classical situation, this is Roiter's theorem. The important point is, that \(n\) can be chosen to be independent of \(M\) and \(N\), which is not so in general, if one considers finitely generated modules over a finitely generated \(R\)-algebra. The author then uses this to get generalizations of results on restricted genera, extension of scalars and cancellation to finitely generated \(\Lambda\)-modules. [Part I, cf. ibid. 18, 169--177 (1984; Zbl 0533.13006).]
finitely generated left \(\Lambda \)-modules
scientific article; zbMATH DE number 4006461
scientific article