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Manis Valuations and Prufer Extensions II
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This volume is a sequel of Manis Valuation and Prüfer Extensions I , LNM1791. The Prüfer 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 well amenable subclass of PM (=Prüfer-Manis) valuations. While in Volume I Prüfer extensions in general and individual PM-valuations have been studied, now the stress 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 Gräter 1980, a far reaching extension of the classical weak approximation theorem in arithmetics. 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, which ultimately goes back to Leopold Kronecker.§Prüfer extensions seem to be mandatory to understand certain aspects of real and p-adic geometry.
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