Manis Valuations And Prufer Extensions Ii (lecture Notes In Mathematics)
by Manfred Knebusch /
2014 / English / PDF
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This volume is a sequel to "Manis Valuation and Prfer Extensions I," LNM1791. The Prfer extensions of a commutative ring A are roughly those commutative ring extensions R / A, where commutative algebra is governed by Manis valuations on R with integral values on A. These valuations then turn out to belong to the particularly amenable subclass of PM (=Prfer-Manis) valuations. While in Volume I Prfer extensions in general and individual PM valuations were studied, now the focus is on families of PM valuations. One highlight is the presentation of a very general and deep approximation theorem for PM valuations, going back to Joachim Grter's work in 1980, a far-reaching extension of the classical weak approximation theorem in arithmetic. Another highlight is a theory of so called "Kronecker extensions," where PM valuations are put to use in arbitrary commutative ring extensions in a way that ultimately goes back to the work of Leopold Kronecker.
