Ideal Theory of Local Quadratic Transforms of Regular Local Rings

Matthew J Toeniskoetter, Purdue University


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.




Heinzer, Purdue University.

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