Applications of linear functionals to Kronecker modules. II
This paper continues the investigation of part I [cf. the preceding review] - a class of modules \(E_ r\), \(r\geq 2\). In this paper a class of modules \(E_ r\), \(r\geq 2\) is constructed from submodules of the module of rational functions K(X), over the polynomial ring K[X], given by a height function h; sequences of linear functionals \(L=(\ell_ 2,...,\ell_ r)\) on K(X); and a sequence of positive integers \((m_ 2,...,m_ r)\). To each subset L' of \(\{\ell_ 2,...,\ell_ r\}\) is attached a field extension \(K_{L'}\) of the prime subfield of K. Let \(M_{L_ 1}\) and \(M_{L_ 2}\) be modules constructed from the sequences of linear functionals \(L_ 1\) and \(L_ 2\). The main results are as follows: If \(M_{L_ 1}\) is isomorphic to \(M_{L_ 2}\), then \(K_{L_ 1}\) and \(K_{L_ 2}\) are contained in finitely generated field extensions of each other. Let \(\bar L_ i=L\setminus \ell_ i\), \(i=2,...,r\). If \(M_ L\) is not purely simple, then for some i in \(\{\) 2,...,r\(\}\), \(K_{\ell_ i}\) is contained in a finitely generated extension of \(K_{\bar L_ i}\). These results are then used to obtain the following theorems: assuming that \(1<\sup p h<\infty\), then there are exactly \(\aleph_ 0\) isomorphism classes of completely decomposable modules in \(E_ r\); there are at least c (the cardinality of the continuum) isomorphism classes of (i) indecomposable but not purely simple modules in \(E_ r\), (ii) purely simple modules in \(E_ r\).
scientific article; zbMATH DE number 3948525
scientific article