On the integral closure of polynomial rings over a valuation domain
Let \(V\) be a valuation domain with fraction field \(K\), \(X\) an indeterminate, \(F\) a finite field extension of \(K(X)\) and \(R\) the integral closure of \(V[X]\) in \(F\). Recall that a domain \(A\) is called an N-2 ring (or Japanese ring), if \(A\) has finite integral closure in any finite field extension of its field of fractions. In the paper under review, the author gives a characterization of the N-2 valuation domains (proving also that \(V\) is N-2 iff so is V[X]). Then he proves that \(R\) is a free \(V[X]\)-module of rank \([F\colon K(X)]\), provided \(R\) is a finite \(V[X]\)-module [this result gives a positive answer to a question posed by \textit{B. Green, M. Matignon} and \textit{F. Pop} in their paper: J. Reine Angew. Math. 432, 117-133 (1992; Zbl 0749.14022)]. Also, the author gives a series of sufficient conditions for the tensor product \(R\otimes_{V} W\) to be a normal domain, where \(W\) is a valuation domain dominating \(V\) and whose fraction field is algebraic over \(K\). In the final section, these results are restated in geometric language.
scientific article; zbMATH DE number 1234244
scientific article