Discrete valuation domains and ranks of their maximal extensions
Let \(R\) denote a valuation domain, \(\overline R\) its completion and \(S\) a maximal immediate extension of \(R\). Let \(\mathrm{Spec}(R)\) be the set of all prime ideals of \(R\), totally ordered by inclusion. Let \(P\in \mathrm{Spec}(R)\). The completion defect \(c_R(P)\) at \(P\) is defined as the rank of the torsionfree \(R/P\)-module \(\overline{R/P}\). The total defect \(d_R(P)\) at \(P\) is the rank of the torsion free \(R/P\)-module \(S/PS\). Some formulae are proved that connect the functions \(c_R\) and \(d_R\), e.g.: Let \(R\) be a discrete valuation domain and \(P\) a prime nonmaximal ideal of \(R\). Then \(d_R(P) = c_R(P) \sup \{d_R(Q); Q\supset P, Q\in \mathrm{Spec}(R)\}\). Finally, the authors completely solve the problem of determining the function ``completion defect'' that can arise from discrete valuation domains \(R\) of non-zero characteristic satisfying the a.c.c. on prime ideals.
scientific article; zbMATH DE number 3983342
scientific article