Directed Unions of Local Quadratic Transforms of a Regular Local Ring

TL;DR

This paper investigates the structure of unions of local quadratic transforms of regular local rings, characterizing when these unions form valuation rings of various ranks and providing new examples and conditions.

Contribution

It extends previous results by analyzing the valuation properties of unions of local quadratic transforms, especially for higher rational ranks and monomial cases.

Findings

Finite sum of valuations v(m_n) when rational rank ≥ 2

Examples of rank 2 valuation domains with non-Z^2 value groups

Conditions for unions to be rank 1 valuation domains

Abstract

We consider the directed union S of an infinite sequence {(R_n, m_n)} of successive local quadratic transforms of a regular local ring (R, m). If dim R = 2, Abhyankar proves that S is a valuation ring. If dim R > 2, Shannon gives necessary and sufficient conditions for S to be a rank 1 valuation domain and Granja gives necessary and sufficient conditions that S be a rank 2 rational rank 2 valuation domain. Granja observes that these are the only cases where S is a valuation domain. If the sequence is along a rank 1 valuation ring V with valuation v, Granja, Martinez, and Rodriguez show that if the infinite sum of the values v(m_n) diverges, then S = V. We prove that this infinite sum is finite if V has rational rank at least 2. We present an example of a sequence whose union S is a rank 2 valuation domain, but whose value group is not Z^2. We also consider sequences of monomial local quadratic transforms and give necessary and sufficient conditions that the union be a rank 1 valuation domain. If it is, it has rational rank d. We string together finite sequences of monomial local quadratic transforms to construct examples where S is a rank 1 valuation domain with rational rank < d.