Integral semihereditary orders, extremality, and Henselization
The paper presents results of the author's Ph.D. thesis. Let \(V\) be a valuation domain with quotient field \(F\), and \(Q\) a central simple \(F\)-algebra. The author proves first that a semi-hereditary \(V\)-order in \(Q\) is extremal, i.e. coincides with the multiplier of its radical. (The converse holds for one-dimensional \(V\)!) Whereas Bézout \(V\)-orders always exist in \(Q\) (e.g. as Dubrovin valuation rings), there need not be semi-hereditary \(V\)-orders. If \(Q\) is a division algebra admitting a total valuation ring extending \(V\), there is a unique semi-hereditary \(V\)-order in \(Q\). If the skewfield part of \(Q\) admits an invariant valuation ring extending \(V\), the semi-hereditary \(V\)-orders are explicitly characterized (as in the classical situation). In this case, the maximal overorders and, in particular, the Dubrovin valuation rings are easily described. The author also shows that the semi-hereditary property is compatible with henselization and proves some pertinent facts in this context.
scientific article; zbMATH DE number 1013984
scientific article