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On relative real holomorphy rings

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Abstract

LetF be a formally real function field (i.e., −1 is not a sum of squares inF) of transcendence degreen over ℝ. LetA be a finitely generated ℝ-subalgebra ofF whose field of quotients isF. The real holomorphy ringH(F|A) ofF overA is the intersection of all valuation rings ofF which containA and have formally real residue field. The ringH(F|A) has been extensively studied and applied by Becker, Schülting and others. It is known to be a Prüfer domain of Krull dimension not exceedingn and from this it can be shown that every finitely generated ideal ofH(F|A) can be generated byn+1 elements. Here, assuming thatA is regular (i.e., every localization with respect to a prime ideal is regular local), we give necessary and sufficient topological conditions in order for every finitely generated ideal ofH(F|A) to admitn generators. We also provide a geometric description ofH(F|A).

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Authors and Affiliations

  1. Department of Mathematics and Statistics, University of New Mexico, 87131, Albuquerque, New Mexico, USA

    M. A. Buchner & W. Kucharz

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  1. M. A. Buchner
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  2. W. Kucharz
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Buchner, M.A., Kucharz, W. On relative real holomorphy rings. Manuscripta Math 63, 303–316 (1989). https://doi.org/10.1007/BF01168373

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