# Ideal Theory of Local Quadratic Transforms of Regular Local Rings

#### Abstract

Let R be a regular local ring of dimension d ≥ 2. To a non-divisorial valuation V that dominates R, there is an associated infinite sequence of local quadratic transforms of R along V. Abhyankar has shown that if d = 2, then the union S of this sequence is equal to V, and in higher dimensions, Shannon and Granja et al. have given equivalent conditions that the union S equals V. In this thesis, we examine properties of the ring S in the case where S is not equal to V. We associate to S a minimal proper Noetherian overring, called the Noetherian hull. Each ring in the sequence has an associated order valuation, and we show that the sequence of order valuations converges to a valuation called the boundary valuation. We show that S is the intersection of its Noetherian hull and boundary valuation ring, and we go on to study these rings in detail. This naturally breaks down into an archimedean and a non-archimedean case, and for each case, we construct an explicit description for the boundary valuation. Then, after loosening the condition that R is regular and replacing the sequence of order valuations with the sequence of Rees valuations of the maximal ideals, we prove an analogous result about the convergence of Rees valuations.

#### Degree

Ph.D.

#### Advisors

Heinzer, Purdue University.

#### Subject Area

Mathematics

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