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arXiv · 1509.07545

Asymptotic properties of infinite directed unions of local quadratic transforms

William Heinzer·Bruce Olberding·Matthew Toeniskoetter
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Abstract

We consider infinite sequences {R_n} of successive local quadratic transforms of a regular local ring. Let S denote the directed union of the sequence of regular local rings R_n. We previously showed the existence of a unique limit point V of the family of order valuation rings of the sequence. In this paper, we examine asymptotic properties of this family of order valuations. We link this asymptotic behavior to ring-theoretic properties of S, namely whether S is archimedean and whether S is completely integrally closed. We construct examples of such S that are archimedean and completely integrally closed but not valuation domains. We give an explicit description of V, where the description depends on whether S is archimedean or non-archimedean.

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William Heinzer, Bruce Olberding, Matthew Toeniskoetter. 2015-09-24. Asymptotic properties of infinite directed unions of local quadratic transforms. https://arxiv.org/abs/1509.07545

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