IDEAL THEORY OF LOCAL QUADRATIC TRANSFORMS OF REGULAR LOCAL RINGS A Dissertation Submitted to the Faculty of Purdue University by Matthew J. Toeniskoetter In Partial Fulfillment of the Requirements for the Degree of Doctor of Philosophy August 2017 Purdue University West Lafayette, Indiana ii THE PURDUE UNIVERSITY GRADUATE SCHOOL STATEMENT OF DISSERTATION APPROVAL Dr. William Heinzer, Chair Department of Mathematics Dr. Bernd Ulrich Department of Mathematics Dr. Giulio Caviglia Department of Mathematics Dr. Edray Goins Department of Mathematics Approved by: Dr. David Goldberg Head of the School Graduate Program iii ACKNOWLEDGMENTS First and foremost, I would like to thank my advisor, Professor William Heinzer, from whom I've learned a great deal. I look back fondly at all the time spent in his office, in classrooms around campus, and in the campus dining halls discussing mathematics. I appreciate all of the support and encouragement he has given me over the years, and while I have not always followed it, I am grateful for all of the helpful advice he has offered. I am thankful to my committee members Professors Bernd Ulrich and Giulio Caviglia, whose graduate level courses have been essential in my understanding of advanced topics in commutative algebra. I am grateful for all of the opportunities to speak at the weekly commutative algebra seminar, which have been invaluable experiences. I am grateful to have Professor Edray Goins on my thesis committee. I thank him for his help both in preparing my thesis defense and in editing and formatting this document. I am also thankful for the opportunity to TA for his research experience for undergraduates program in the summer of my third year in graduate school. I would like to thank my collaborators Youngsu Kim, Mee-Kyoung Kim, Alan Loper, Guerrieri Lorenzo, Bruce Olberding, and Hans Schoutens. I had a great time working with all of you. In addition, I would like to thank Alan for the invitation to speak at the Ohio State University commutative algebra seminar, and Bruce, for his contributions to the project that comprises much of my thesis. I am thankful to the undergraduate professors involved in the research for under- graduates program at Oakland University: Eddie Cheng, Meir Shillor, and especially Anna Spagnuolo, who sent me on the path to Purdue University. I would like to thank the mathematics department at Purdue University for pro- viding me with the wonderful opportunity to pursue the study of mathematics. iv Finally, I thank my parents. I am grateful for all of their patience and support throughout my study of mathematics. Matthew Toeniskoetter West Lafayette, July 19, 2017 v TABLE OF CONTENTS Page ABSTRACT ::::::::::::::::::::::::::::::::::::: vii 1 INTRODUCTION :::::::::::::::::::::::::::::::: 1 2 PRELIMINARIES :::::::::::::::::::::::::::::::: 7 2.1 Notation and Definitions :::::::::::::::::::::::::: 7 2.2 The Dimension Formula :::::::::::::::::::::::::: 7 2.3 Ordered Abelian Groups :::::::::::::::::::::::::: 8 2.4 Valuations and Valuation Rings :::::::::::::::::::::: 9 2.5 Transforms of Ideals :::::::::::::::::::::::::::: 11 3 CONVERGENCE OF VALUATIONS :::::::::::::::::::::: 13 3.1 Valuation Rings as a Topological Space :::::::::::::::::: 13 3.2 Limit of Valuation Rings :::::::::::::::::::::::::: 14 3.3 Possibly Infinite Valuations :::::::::::::::::::::::: 15 4 SEQUENCES OF LOCAL QUADRATIC TRANSFORMS OF REGULAR LOCAL RINGS :::::::::::::::::::::::::::::::::: 18 4.1 Blow-Ups and Local Quadratic Transforms :::::::::::::::: 18 4.2 Local Quadratic Transforms and Transforms of Ideals :::::::::: 20 4.3 Quadratic Shannon Extensions :::::::::::::::::::::: 21 4.4 The Noetherian Hull :::::::::::::::::::::::::::: 25 4.5 The Boundary Valuation :::::::::::::::::::::::::: 26 5 THE STRUCTURE OF A SHANNON EXTENSION ::::::::::::: 28 5.1 The Intersection Decomposition :::::::::::::::::::::: 29 5.2 The Asymptotic Limit of Order Valuations :::::::::::::::: 30 5.3 The Limit of Transforms :::::::::::::::::::::::::: 33 5.4 The Transform Formula for w ::::::::::::::::::::::: 34 5.5 The Archimedean Case ::::::::::::::::::::::::::: 37 5.6 The Complete Integral Closure :::::::::::::::::::::: 41 5.7 The Non-Archimedean Case :::::::::::::::::::::::: 43 5.8 Completely Integrally Closed Examples :::::::::::::::::: 46 6 LIMITS OF REES VALUATIONS ::::::::::::::::::::::: 50 6.1 Normalizing Local Quadratic Transforms ::::::::::::::::: 50 6.2 Rees Valuations ::::::::::::::::::::::::::::::: 51 6.3 Multiplicity and Blowing Up :::::::::::::::::::::::: 53 6.4 Degree Functions :::::::::::::::::::::::::::::: 55 vi Page 6.5 Convergence of Rees Valuations :::::::::::::::::::::: 57 7 MONOMIAL LOCAL QUADRATIC TRANSFORMS ::::::::::::: 60 7.1 Monomial Valuations :::::::::::::::::::::::::::: 61 7.2 Conditions for the union to be a valuation :::::::::::::::: 63 REFERENCES :::::::::::::::::::::::::::::::::::: 68 vii ABSTRACT Toeniskoetter, Matthew J. PhD, Purdue University, August 2017. Ideal Theory of Local Quadratic Transforms of Regular Local Rings. Major Professor: William Heinzer. 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 fRngn≥0 of R along V . Abhyankar has shown that if d = 2, then the union S = S n≥0 Rn 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 Rn has an associated order valuation ordRn , and we show that the sequence of order valuations fordRn gn≥0 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 fordRn gn≥0 with the sequence of Rees valuations fνngn≥0 of the maximal ideals of Rn, we prove an analogous result about the convergence of Rees valuations. 1 1. INTRODUCTION The notion of blowing up is a powerful tool that arises in the study of commutative algebra and algebraic geometry. Speaking loosely and geometrically, the notion of blowing up is to replace a subspace of a space with a larger space that can distinguish the different directions through the original subspace. For instance, the curve x2 − y2 + x3 = 0 takes two distinct paths through the origin. By blowing up the origin in the plane, we replace the origin with a projective line. Then after taking the strict transform of the curve to the blown-up space, the two distinct paths that the curve takes through the origin now intersect this projective line in two distinct points and the curve no longer intersects itself. Algebraically, to blow-up the ideal I of the Noetherian ring R, one uses the Rees algebra of I, defined as M R[It] = Intn = R ⊕ It ⊕ I2t2 ⊕ · · · : n≥0 The blow-up of I is the projective scheme ProjR[It]. Now let R be a regular local ring of dimension d ≥ 2 and let m = (x1; : : : ; xd) be the maximal ideal of R. Take the blow-up of m, look at the local ring of a point in the fiber of m, and call this ring R1. The ring R1 is a local quadratic transform of R. Since mR1 is principal, mR1 = xkR1 for some k with 1 ≤ k ≤ d. Algebraically, the ring R1 has the form x1 xd R1 = R ; ··· ; xk xk n for some prime ideal n such that xk 2 n. It is well-known that the ring R1 is itself a regular local ring and the dimension of R1 is at most the dimension of R. One may iterate this process, and in this way, obtain a sequence f(Rn; mn)gn≥0 of local quadratic transforms of regular local rings. The dimensions of the rings in this sequence are non-increasing positive integers, so the dimension must stabilize. 2 This process of iterating local quadratic transforms of the same dimension corre- sponds to the geometric notion of tracking nonsingular closed points through repeated blow-ups. We can think of the ring R itself as some nonsingular closed point on some variety. In taking a local quadratic transform, one blows up the closed point R on this variety, then chooses a closed point R1 in the fiber of R. An instance where such a sequence naturally arises is through a valuation ring V that birationally dominates R = R0. The valuation ring V has a unique center on the blow-up ProjR[mt] of m, which uniquely determines the local quadratic transform of R along V . This iterative geometric process is a powerful tool, especially in a two-dimensional setting. It plays a central role in the embedded resolution of singularities for curves on surfaces (see, for example, [2] and [5, Sections 3.4 and 3.5]), as well as factorization of birational morphisms between nonsingular surfaces (see [1, Theorem 3] and [28, Lemma, p. 538]). Algebraically, iterated local quadratic transforms are an essential component in Zariski's theory of integrally closed ideals in two-dimensional regular local rings (see [29, Appendix 5]). Every such ideal has a unique factorization into irreducible such ideals, and the irreducible ideals are in one-to-one correspondence with the two-dimensional regular local rings that can be attained through iterated local quadratic transforms. Abhyankar proves that for a sequence fRngn≥0 of iterated local quadratic trans- S forms of two-dimensional regular local rings, the union Rn is a valuation ring [1, n≥0 Lemma 12]. For rings of higher dimension, this is no longer true. Shannon gives ex- S amples [27, Examples 4.7 and 4.17] that show that the union S = Rn of a sequence n≥0 of local quadratic transforms of three-dimensional regular local rings need not be a valuation ring. In the same paper, Shannon proves an equivalent condition for the union S to be a rank 1 valuation ring.