IDEAL THEORY OF LOCAL QUADRATIC TRANSFORMS OF REGULAR LOCAL RINGS. A Dissertation. Submitted to the Faculty. Purdue University

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 undergraduates 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 providing 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 {R n } n 0 of R along V. Abhyankar has shown that if d = 2, then the union S = n 0 R n 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 R n has an associated order valuation ord Rn, and we show that the sequence of order valuations {ord Rn } n 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 {ord Rn } n 0 with the sequence of Rees valuations {ν n } n 0 of the maximal ideals of R n, 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 x 2 y 2 + x 3 = 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 R[It] = I n t n = R It I 2 t 2. 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 = (x 1,..., x d ) 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 R 1. The ring R 1 is a local quadratic transform of R. Since mr 1 is principal, mr 1 = x k R 1 for some k with 1 k d. Algebraically, the ring R 1 has the form [ x1 R 1 = R,, x ] d x k x k n for some prime ideal n such that x k n. It is well-known that the ring R 1 is itself a regular local ring and the dimension of R 1 is at most the dimension of R. One may iterate this process, and in this way, obtain a sequence {(R n, m n )} n 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 corresponds 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 R 1 in the fiber of R. An instance where such a sequence naturally arises is through a valuation ring V that birationally dominates R = R 0. 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 {R n } n 0 of iterated local quadratic transforms of two-dimensional regular local rings, the union R n is a valuation ring [1, Lemma 12]. For rings of higher dimension, this is no longer true. Shannon gives examples [27, Examples 4.7 and 4.17] that show that the union S = R n of a sequence 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. In recent papers, Granja et al. find equivalent conditions for S to be a rank 2 valuation ring, and go on to show that these are all the cases where S is a valuation ring. n 0 n 0

3 We consider the case where S is not a valuation ring, and we analyze the structure of the ring S and the properties of the sequence {R n } n 0 from which S arises. In honor of David Shannon s work, we call S a quadratic Shannon extension of the ring R. The ring S is local and integrally closed, but it is Noetherian only in the case where S is a discrete valuation ring. However, while S itself is generally not Noetherian, its punctured spectrum is the spectrum of a Noetherian ring called the Noetherian hull of S. The Noetherian hull, which we denote by T, can be explicitly constructed as T = S[ 1 x ] for any m S-primary element x m S. It is the minimal proper Noetherian overring of S. Each of the rings R n has an order valuation associated to its maximal ideal, denoted by ord Rn. The sequence of order valuations converges, in a topological sense, to a valuation that birationally dominates S. This valuation is called the boundary valuation of the sequence, and we denote it by V. Whereas the Noetherian hull T is implicit to the ring S and does not depend on the sequence, the boundary valuation ring V arises not from S but from the sequence itself. We prove in Theorem 5.1.1 that S is the intersection of the Noetherian hull and the boundary valuation ring. Theorem 5.1.1. Let S be a Shannon extension and let T and V denote its Noetherian hull and boundary valuation ring, respectively. Then S = T V. Next, we study the boundary valuation itself through two limits e( ) and w( ) that arise from the sequence of order valuations. For a principal ideal ar 0, we consider its sequence of successive transforms, {(ar 0 ) Rn } n 0. The corresponding sequence of orders {ord Rn ((ar 0 ) Rn )} n 0 eventually stabilizes, and we denote the limit by e(a). The function e( ) is multiplicative, but it is not always itself a valuation. To describe w, we fix an arbitrary m S -primary element x m S to set w(x) = 1. Then w is defined by w(a) = lim n ord Rn (a) ord Rn (x).

4 We show in Theorem 5.2.1 that this limit exists, but it may be ± for some nonzero choices of a. While w( ) sometimes takes non-finite values, it is always itself a valuation in some sense. Shannon extensions naturally break down into two cases: an archimedean case and a non-archimedean case. To distinguish the two cases, we introduce the invariant τ, defined by τ = w(m n ). n=0 In Theorem 5.5.4, we give equivalent conditions for S to be archimedean. Namely, the following are equivalent: 1. S is archimedean. 2. w(a) is finite for all a 0. 3. τ <. 4. T is not local. In Theorem 5.5.2, we describe explicitly the boundary valuation as the direct sum of w and e, ordered lexicgraphically. An element has positive V -value if and only if either it has positive w-value, or it has zero w-value and negative e-value. In particular, w determines the unique rank 1 overring of boundary valuation ring V and V has either rank 1 or rank 2. We go on to show in Theorem 5.6.2 that V has rank 1 if and only if S is completely integrally closed. The example [27, Example 4.17] given by Shannon is archimedean and not completely integrally closed. We construct an explicit Shannon extension that is both archimedean and completely integrally closed in Example 5.8.3. Its boundary valuation has rank 1 and rational rank 2. In the non-archimedean case, S has a unique dimension 1 prime ideal p and T = S p. In Theorem 5.7.2, we given an explicit construction of the boundary valuation. Theorem 5.7.2. Let S be a non-archimedean Shannon extension.

5 1. e( ) is a rank 1 discrete valuation that defines a DVR (E, m E ). 2. w( ) induces a rank 1 rational rank 1 valuation w( ) that defines a valuation ring W on the residue field E/m E. 3. V is the composite valuation of E and W. In the non-archimedean case, the boundary valuation always has rank 2 and rational rank 2. The situation is the reverse of the archimedean case. In the archimedean case, V is defined by considering w first, then e second, but for the non-archimedean case, V is defined by considering e first, then w second. In Chapter 6, we give a partial generalization. We relax the condition that the ring R be regular and consider a normal Noetherian local domain R. Whereas the blowup of a regular ring is always regular, the blow-up of a normal Noetherian domain need not be normal, so we take its integral closure in the notion of a local normalized quadratic transform. In addition, to ensure that the integral closure is still Noetherian, we must impose the mild geometric condition that R be analytically unramified. Then, as in the case for regular local rings, we consider sequences {(R n, m n )} n 0 of local normalized quadratic transforms of analytically unramified Noetherian local domains. As in the case of regular local rings, we associate a sequence of valuations. To an ideal I of a Noetherian domain, there is an associated set of discrete valuations called the Rees valuations of I that naturally arise through the blow-up. For a regular local ring (R, m), the Rees valuation of m is precisely the order valuation ord R. By making the assumption that the ideals m n have a unique Rees valuation ν n, i.e. that they are one fibered, we obtain a sequence {ν n } n 0 of valuations. We prove in Theorem 6.5.2 that, as in the case of order valuations of regular local rings, the sequence {ν n } n 0 of Rees valuations converges. Theorem 6.5.2. Let {(R n, m n )} n 0 be a sequence of local normalized quadratic transforms of analytically unramified Noetherian local domains. If m n has a unique

6 Rees valuation ν n for each n 0, then the sequence {ν n } n 0 of Rees valuations converges in the patch topology.

7 2. PRELIMINARIES In this chapter, we fix notation and review some background material necessary for later chapters. 2.1 Notation and Definitions We follow the notation as in [23]. All rings are commutative with identity. For a ring R, denote by dim R the Krull dimension of R. A local ring is a ring R with a unique maximal ideal, denoted by m R unless otherwise specified, and need not be Noetherian. Let A be a domain and denote by Q(A) the field of fractions of A. Let A B be an inclusion of rings. The ring B is birational over A, and B is an overring of A, if A B Q(A). Let (R, m R ) and (S, m S ) be local rings. A ring homomorphism φ : R S is called local if φ 1 (m S ) = m R. For an inclusion of local rings R S, the ring S dominates R if the inclusion map is local, i.e. if m S R = m R. Let R be a ring and M an R-module. We denote by Ass(M) and Min(M) the associated primes and minimal primes of M, respectively. 2.2 The Dimension Formula Let (R, m R ) be a Noetherian local domain and let (S, m S ) be a local domain that dominates R. Then the following formula, called the dimension inequality [23, Theorem 15.5], holds: dim S + tr. deg. R/mR (S/m S ) dim R + tr. deg. Q(R) Q(S). (2.1)

8 A Noetherian ring R is called catenary if for every inclusion of prime ideals P Q of R, every maximal chain of prime ideals P = P 0 P 1... P n = Q from P to Q has the same length. The ring R is called universally catenary if every finitely generated R-algebra is catenary. Remark 2.2.1. Virtually every ring that arises in algebraic geometry is universally catenary. The following are examples of universally catenary rings: 1. Essentially finitely generated algebras over a field. 2. Cohen-Macaulay rings. 3. Complete Noetherian local rings. If the Noetherian local domain R as in the dimension inequality is universally catenary, and if the local domain S that dominates R is essentially finitely generated over R, then the dimension inequality becomes an equality, the dimension formula [23, Theorem 15.6]: dim S + tr. deg. R/mR (S/m S ) = dim R + tr. deg. Q(R) Q(S). (2.2) 2.3 Ordered Abelian Groups Let Γ be a linearly ordered abelian group. Define an equivalence relation on Γ >0 by x y if and only if there exists positive integers p and q such that px > y and qy > x. The equivalence classes of are themselves linearly ordered, where if [x] [y], then [x] < [y] if and only if x y. The cardinality of the set of equivalence classes of Γ is called the rank of Γ and is denoted rank Γ. We recall the Hahn embedding theorem [11]. Let Ω denote the equivalence classes of. Then there exists an embedding of Γ into R Ω, where R Ω is the linearly ordered

9 abelian group of all real-valued functions from Ω to R that vanish outside of a wellordered set. A set S of elements of Γ is called rationally independent if for any Z-linear relation p 1 x 1 +... + p n x n = 0 where x 1,..., x n are distinct elements in S and p 1,..., p n Z, then p 1,..., p n = 0. The cardinality of a maximal rationally independent set is the rational rank of Γ and is denoted r. rank Γ. By choosing a representative from each equivalence class of, it follows that rank Γ r. rank Γ. 2.4 Valuations and Valuation Rings A valuation on a field K is a map ν : K G, where G is some linearly ordered abelian group, that satisfies the following properties for all x, y K : 1. ν(1) = 0. 2. ν(xy) = ν(x) + ν(y). 3. ν(x + y) min{ν(x), ν(y)}. The valuation ν is non-trivial if ν(k ) {0}. By convention, we denote ν(0) = +. The linearly ordered abelian group ν(k ) G is called the value group of ν. Associated to a valuation ν is the valuation ring V of ν defined as the set of elements of K with nonnegative ν-value, V = {x K ν(x) 0}. The ring V is local and its maximal ideal m V is the set of elements K with positive ν-value, m V = {x K ν(x) > 0}. The valuation ring of the trivial valuation is the field K itself. We recall equivalent definitions for a domain to be a valuation ring:

10 Theorem 2.4.1. Let V be a domain with field of fractions K. The following are equivalent: 1. V is the valuation ring of some valuation on K. 2. For all nonzero x K, either x V or x 1 V. 3. V is local and V is maximal with respect to birational domination. Let ν be a valuation with value group Γ and valuation ring V. The rank of ν is the rank of its value group rank ν = rank Γ and the rational rank of ν is the rational rank of its value group r. rank ν = r. rank Γ. Moreover, rank ν = dim V ; the equivalence classes of are in one-to-one correspondence with the prime ideals of V. In our applications, we consider valuation rings that are overrings of a Noetherian domain A. The dimensional inequality Equation 2.1 implies that dim V dim A mv A, which implies that V has finite rank. By the Hahn embedding theorem, there exists an embedding Γ R rank ν, a finite number of copies of R ordered lexicographically. Moreover, Abhyankar proves a formula analogous to the dimension inequality Equation 2.1 involving the rational rank of ν. For a valuation ring V birationally dominating a Noetherian local domain (R, m), we have, by [1, Proposition 2], tr. deg. R/m V/m V + r. rank ν dim R. (2.3) Moreover, if Equation 2.3 is an equality, then the value group of ν is isomorphic as an (unordered) abelian group to Z dim R and the residual field extension R/m V/m V is finitely generated. Such valuations are called Abhyankar valuations. A valuation ν whose value group is a subgroup of R is called a rank 1 real valuation. By the Hahn embedding theorem, every rank 1 valuation can be written as a rank 1 real valuation. A valuation ν is called discrete if its value group is of the form Z rank ν. A rank 1 discrete valuation is called a Discrete Valuation Ring (DVR) and has special importance in commutative algebra and algebraic geometry. We recall equivalent conditions for a local ring to be a DVR:

11 Theorem 2.4.2. equivalent: [23, Theorem 11.2] Let (R, m) be a local ring. The following are 1. R is a 1-dimensional integrally closed Noetherian local domain. 2. R is a Noetherian ring and m is principal. 3. R is a Noetherian valuation ring. 4. R is a DVR. 2.5 Transforms of Ideals Given a UFD A with field of fractions K and an A-ideal I A, there exists a unique representation I = gcd(i)j, where J is an A-ideal of height 2. The principal ideal gcd(i) can be written in the form P α 1 1 P αn, unique up to re-ordering, where P 1,..., P n are height 1 prime ideals of A. We recall some preliminary material from [22]. Consider a pair of UFDs A B such that B is birational over A. Let P be a height 1 prime ideal of A. Then either the DVR A P contains B, in which case P A P B is a height 1 prime ideal and A P = B P AP B, or A P does not contain B. The transform of the height 1 prime ideal P in B is defined, depending on these two cases, to be P A P B if B A P, P B = B otherwise. For an ideal J A of height 2, the transform of J in B is defined to be J B = n JB gcd(jb). For general ideals I A, the transform of I in B is defined multiplicatively. Writing I uniquely in the form I = P 1 P n J, where htj 2, the transform of I in B is defined to be I B = P B 1 P B n J B.

12 Essentially, by taking the transform of an ideal of A in B, we are extending the ideal, then deleting the principal prime factors that appear in B but do not exist in A. We recall basic facts about transforms. Taking the transform of an ideal in the same ring has no effect, taking transforms is multiplicative, and taking transforms is transitive. Remark 2.5.1. 1. If A is a UFD and I A is an ideal, then I A = I. 2. If A B is a birational inclusion of UFDs and I, J A are ideals, then (IJ) B = I B J B. 3. If A B C are birational inclusions of UFDs and I A is an ideal, then (I B ) C = I C. Since a fractional A-ideal I K has the same unique representation as a genuine A-ideal, but with possibly negative exponents to the height 1 factors, the same notion makes sense for fractional A-ideals. Given a fractional A-ideal I, we may write I = P α 1 1 P αn J uniquely, where P 1,..., P n are distinct height 1 primes of A, α 1,..., α n n are nonzero (but possibly negative) integers, and J is an A-ideal of height 2. Then I B = (P1 B ) α1 (Pn B ) αn J B is the transform of the fractional A-ideal I in B, and is itself a fractional B-ideal. Transforms of fractional ideals satisfy all of the same properties as in the previous remark.

13 3. CONVERGENCE OF VALUATIONS 3.1 Valuation Rings as a Topological Space For this section, we fix the following notation: Setting 3.1.1. Let R be a Noetherian domain with field of fractions K and let V denote the collection of valuation overrings of R. The collection V is naturally endowed with the Zariski topology. For a finite subset A K, there is a corresponding open set in the Zariski topology, U A = {V V A V }. These sets U A form an open basis for the Zariski topology. Notice that V \ U {x 1 } = {V V x 1 / V } = {V V x m V }. The patch topology on V, also called the constructible topology, is a refinement of the Zariski topology. The patch topology is defined as the coarsest topology such that every open set and every closed set in the Zariski topology are open in the patch topology. For A, B K finite sets, we define the open set U A,B = U A (V \ U B ) = {V V A V, B m V }. These sets U A,B form an open basis for the patch topology. The Zariski topology on V is very coarse. It has a generic point K, where {K} = V. The Zariski topology on V is compact, but it is never Hausdorff. The patch topology on V is finer, and it is both compact and Hausdorff.

14 3.2 Limit of Valuation Rings In this section, we introduce the notions of convergence of valuations and valuation rings with respect to the patch topology. Let {V n } n 0 be a sequence of valuation rings. Since the patch topology on the space of valuation rings is Hausdorff, the notion of convergence is a sensible one; if the sequence {V n } n 0 is convergent in a topological sense, then it has a unique limit point lim n V n. Moreover, since the patch topology is compact, every infinite set of valuation overrings of R has at least one limit point. However, the patch topology is generally not sequentially compact. While the sequence {V n } n 0 has at least one limit point, it is possible that no subsequence of {V n } n 0 is convergent. We translate the topological condition of convergence to an algebraic one: Definition 3.2.1. Let {V n } n 0 be a sequence of valuation rings. This sequence is convergent if and only if A = n 0 is a valuation ring. In this case, the sequence converges to the valuation ring A. m n V m We restate the definition in terms of valuations: Definition 3.2.2. Let {ν n } n 0 be a sequence of valuations. This sequence is convergent if and only if for every x K, there exists N large such that the sign of ν n (x) is constant for n N. We concern ourselves primarily with sequences of valuation rings that dominate an ascending sequence of Noetherian local domains. In the following remark, we show that if the union of the Noetherian local domains is a valuation ring, then any such sequence of valuation rings converges to it. Remark 3.2.3. Let {R n, m n } n 0 be an infinite birational dominating sequence of Noetherian local domains (i.e. m n+1 R n = m n ) whose union R n is a valuation domain V. For each n 0, let V n be a valuation ring that birationally dominates R n. Then the sequence of valuation rings {V n } n 0 converges to V. n 0

15 Proof. For x V, it follows that x R n for n 0, so x V n for n 0. For x / V, since x 1 V, it follows that x 1 R n for n 0, so x 1 m n for n 0. Since V n dominates R n, it follows that if x 1 m Vn for n 0, i.e. x / V n for n 0. 3.3 Possibly Infinite Valuations To analyze the limit of a sequence of valuations, we introduce the notion of a possibly infinite valuation, or p.i. valuation. In essence, a p.i. valuation on a field K is a valuation that allows nonzero elements of K to have value ±. Definition 3.3.1. Let R be an integral domain with field of fractions K. A possibly infinite valuation ν on K is a function ν : R Γ {± } where Γ is an ordered abelian group and ν satisfies the ordinary conditions for a valuation, 1. ν(xy) = ν(x) + ν(y), unless this sum is of the form ±( ). 2. ν(x + y) min{ν(x), ν(y)}. 3. ν(1) = 0 and ν(0) =. The p.i. valuations induce valuations rings in the same way that ordinary valuations do. Indeed, the set of valuation rings associated to p.i. valuations is the same as the set of valuation rings associated to ordinary valuations. Remark 3.3.2. Let R be an integral domain with field of fractions K and let ν be a p.i. valuation on K. Then: V = {x K ν(x) 0} is a valuation ring with maximal ideal m V = {x K ν(x) > 0}.

16 p = {x R ν(x) = + } is a prime ideal of R called the support of ν. ν induces an ordinary valuation on the field of fractions of R/p. In the following remark, we compare different notions of valuations. Remark 3.3.3. In the usual definition of valuation, ν(x) = only for the value x = 0, and ν extends naturally from R to a function on K. In the definition of a valuation given by Bourbaki [4, VI, 3.2], nonzero elements x of R may have value ν(x) =. A valuation in the sense of Bourbaki has a similarly defined support p and similarly induces an ordinary valuation on the field of fractions of R/p. However, in Bourbaki s definition, the function ν is defined only on R and does not always extend uniquely to a function on K. This is because for nonzero elements x, y R such that ν(x) = ν(y) =, the value of x y is ambiguous. We assume in the definition of a p.i. valuation that ν is already defined on all of K, removing this ambiguity. The p.i. valuations arise in the context of limits of valuations, and in particular the limits of real valuations. Let {ν n } n 0 be a convergent sequence of real valuations. By choosing a nonunit x, fixing its value to be 1, and taking the asymptotic limit, one obtains a p.i. valuation describing the limit. Theorem 3.3.4. Let R be a local domain with field of fractions K. Let {ν n } n 0 be a sequence of rank 1 valuations on K dominating R that converges to a valuation ring V. Then for nonzero x m R, the limit ω x (y) = lim n ν n (y) ν n (x) exists for all y K, but may be ± for nonzero values of y. is a possibly infinite valuation whose valuation ring V ωx The function ω x is a valuation overring of V. Moreover, xv ωx is m Vωx -primary; in particular, V ωx = V if and only if xv is m V -primary.

17 Proof. Fix a nonzero y K. Let p q Q be any rational number. Since the sequence ν n converges, the sign of ν n ( yp x q ) is constant for n 0. Rewriting, the sign of pν n (y) qν n (x) is constant for n 0. That is, either νn(y) ν n(x) < p q for n 0, ν n(y) ν n(x) = p q for n 0, or νn(y) ν n(x) > p q for n 0. Thus every rational number is an eventual upper or lower bound for the sequence { νn(y) ν n(x) } n 0, so the sequence either converges to some real number or diverges to ±. Thus ω x (y) exists for all y K. To verify that ω x is a p.i. valuation, let y, z K. The sequence { νn(yz) ν n(x) } n 0 equals { νn(y) ν n(x) + νn(z) ν n(x) } n 0, so ω x (yz) = ω x (y)+ω x (z) unless this expression is ±( ). The sequence { νn(y+z) } ν n(x) n 0 is termwise greater than or equal to {min{ νn(y), νn(z) }} ν n(x) ν n(x) n 0, so ω x (y + z) min{ω x (y), ω x (z)}. Thus ω x is a p.i. valuation. To see that the valuation ring V ωx is a valuation overring of V, let y V. Then ν n (y) 0 for n 0 by definition of V. Since x m R and each of the ν n dominate R, it follows that ν n (x) > 0 for all n 0. Therefore νn(y) ν n(x) 0 for n 0, so ω x(y) 0 and y V ωx. Thus V ωx is a valuation overring of V. Since x has positive finite value in V ωx, it is m Vωx -primary. Since V ωx is a valuation overring of V, we conclude that V ωx = V xv, and in particular V = V ω if and only if xv = mv. While each valuation in the sequence of valuations in Theorem 3.3.4 has rank 1, and the p.i. valuation obtained as the limit ω x maps to R {± }, the valuation ring corresponding to ω x can have arbitrary rank. The following example shows how one can construct a maximal rank valuation over a polynomial ring. Example 3.3.5. Let R = k[x 1,..., x d ] be a polynomial ring in d variables over a field k. For n 0, let ν n be the real monomial valuation such that ν n (x i ) = n i. Then the sequence {ν n } n 0 converges to a rank d valuation ring described by the valuation ν : R Z d, where Z d is ordered lexicographically. The valuation ν is defined by ν(x i ) = (0,..., 1,..., 0), where the 1 is in the (d i + 1)-th position, so that ν n (x d ) = (1, 0,..., 0) has the largest value and ν n (x 1 ) = (0,..., 0, 1) has the smallest value. The p.i. valuation ω xi has rank d i + 1.

18 4. SEQUENCES OF LOCAL QUADRATIC TRANSFORMS OF REGULAR LOCAL RINGS 4.1 Blow-Ups and Local Quadratic Transforms Let (R, m) be a Noetherian local domain, let I be a nonzero proper ideal of R, and let V be a valuation domain birationally dominating R. Then one can construct the blow-up or dilatation of I along V, say (S, n). The ring S is the minimal local overring of R dominated by V such that IV is principal. It is realized ring-theoretically by selecting an element a I such that IV = av, then localizing the ring R[ I ] at the a center of V. The element a in this construction is not uniquely determined, but the ring S is uniquely determined after localization. Geometrically, the blow-up of I along V is the stalk at the center of V on the blow-up ProjR[It] of I. If I is already principal for instance, if R is a DVR then S = R. Otherwise, S is a proper overring of R. Since S is essentially finitely generated over R, it is itself a Noetherian local domain dominated by V. The dimension inequality Equation 2.1 implies that dim S dim R. The blow-up of I is an isomorphism on the spectra outside of the fiber of I, SpecR \ V(I) ProjR[It] \ V(IR[It]). Let p SpecR be a prime ideal such that I p. If there exists a prime ideal q SpecS such that q R = p, then R p = S q. The blow-up of the maximal ideal m of R along V has a special name: it is the local quadratic transform of R along V. Denote (R 0, m 0 ) = (R, m). By iterating local quadratic transforms, there exists a birational local sequence {(R n, m n )} n 0 of Noetherian local domains, where (R n+1, m n+1 ) is the local quadratic transform of

19 (R n, m n ) along V. Set (R n+1, m n+1 ) to be the local quadratic transform of (R n, m n ) along V, so R = R 0 R 1 R n 1 R n. If R n+1 = R n for any n 0, or equivalently, if m n is principal for any n 0, then R n is a DVR by Theorem 2.4.2. Since valuation domains are maximal with respect to domination by Theorem 2.4.1, it follows that V = R n and the sequence stabilizes after a finite number of steps. If V R n for any n 0, then R n R n+1 for all n 0. Thus the sequence {(R n, m n )} n 0 of local quadratic transforms is an infinite sequence of proper Noetherian overrings. Assume that R = R 0 is universally catenary, so the dimension formula Equation 2.2 holds. Suppose that the sequence of local quadratic transforms along V stabilizes with R n = V, so V is a DVR. Then the dimension formula Equation 2.2 implies that V is a divisorial valuation ring. Thus if a valuation ring V is not divisorial, then the sequence {(R n, m n )} n 0 of local quadratic transforms along V does not stabilize. An important special case of local quadratic transforms is for regular local rings (RLRs), which are always universally catenary [23, Theorem 17.9]. A 1-dimensional RLR is already a DVR, so assume dim R 0 2. If (R 0, m 0 ) is an RLR and (R 1, m 1 ) is a local quadratic of m, then R 1 is itself an RLR [1, Lemma 10]. Let {(R n, m n )} n 0 be the sequence of local quadratic transforms along a valuation V birationally dominating R 0. An RLR R n of dimension 2 in that appears in any such sequence is an infinitely near point of R 0, denoted R 0 R n. The relation is a partial order, where R S if S is an RLR of dimension 2 that is obtained by an iterated sequence of local quadratic transforms of R. Abhyankar proves in [1, Proposition 3] that an infinite sequence of local quadratic transforms of regular local rings along V stabilizes if and only if V is a divisorial, and furthermore, if this sequence stabilizes with R n = V, then V is the order valuation ring of R n 1. Thus there is a one-to-one correspondence {order valuations of rings S R } {divisorial valuations dominating R}.

20 4.2 Local Quadratic Transforms and Transforms of Ideals Let (R, m) be a regular local ring and let R be a local quadratic transform of R. Since the rings R and R are UFDs, we have the notion of the transform of an R-ideal or a fractional R-ideal from R to R as in Section 2.2.5. Let J K be such a fractional ideal. Then the transform J R of the fractional R-ideal J in the ring R is the ideal given the by equation m ord R(J) J R = JR. (4.1) If J is an R-ideal, then its order is nonnegative, so the exponent of m is nonnegative. If J is a fractional R-ideal with negative order, it still makes sense for m to have a negative exponent, as mr is principal. mr = xr, we may rewrite this equation as J R = By fixing an element x m such that JR x ord R(J). (4.2) The order of an R-ideal J R is non-increasing under transform. We record this in the following remark. Remark 4.2.1. Let (R, m) be a regular local ring and let R be a local quadratic transform of R. If J R is an R-ideal, then ord R (J R ) ord R (J). For an iterated sequence of local quadratic transforms, we can iterate the process of taking the transform of an ideal. Let {(R n, m n )} m n=0 be a finite sequence of local quadratic transforms and let J be a fractional R 0 -ideal. Then [ m 1 ] n=0 m ord Rn (J Rn ) n J Rm = JR m. (4.3) As before, if J is an R-ideal, then each exponent is nonnegative, but the notion of negative exponents is still sensible for fractional R-ideals. By fixing elements x n m n such that m n R m = x n R m for 0 n < m, we may rewrite this equation as J Rm = m 1 n=0 JR m x ord Rn (J Rn ) n. (4.4)

21 Given an infinite sequence {(R n, m n )} n 0 of local quadratic transforms, and an R 0 -ideal J R 0, then the sequence {ord Rn (J Rn )} n 0 is a non-increasing sequence of nonnegative integers, so it eventually stabilizes. Since any fractional R 0 -ideal J K can be written in the form J = a 1 J for some a R 0 and some R 0 -ideal J R 0, and as transforms of ideals are multiplicative, it follows that the sequence {ord Rn (J Rn )} n 0, while it may fluctuate finitely many times, also eventually stabilizes. 4.3 Quadratic Shannon Extensions Shannon considered in [27] and Granja et al. considered in [6], [7], [8] the behavior of infinite sequences of local quadratic transforms and the structure of the directed union. We formalize the notation in the following setting to discuss their results. Setting 4.3.1. Let {(R n, m n )} n 0 be an infinite sequence of local quadratic transforms of regular local rings of dimension 2, R 0 R 1 R n. Denote S = R n and m S = m n. n 0 n 0 We call S the (quadratic) Shannon extension of this sequence in honor of David Shannon s foundational work on these sequences. The directed union S is the union of integrally closed local rings, so it is itself integrally closed and local with maximal ideal m S. However, unless S is a DVR, it is not Noetherian. Remark 4.3.2. [16, Corollary 3.9] If S is Noetherian, then S is a DVR. Proof. For each n 0, fix x n m n such that m n R n+1 = x n R n+1, and notice that x n 1 R n+1 m n R n+1 = x n R n+1. Hence x 0 S x 1 S... is an infinite ascending chain of ideals in S. If S is Noetherian, then this chain stabilizes, so there exists k such that x k S = (x 1, x 2,...)S. But m S = m n = (x 1, x 2,...)S = x k S, so m S is principal. Since S is a Noetherian local domain whose maximal ideal is principal, S is a DVR [23, Theorem 11.2]. n 0

22 Since S is integrally closed, if S is not itself a valuation domain, then there exists a valuation ring V that birationally dominates S with positive residual transcendence degree, i.e. such that tr. deg. S/mS (V/m V ) > 0 [29, Ch. IV 5 Theorem 10]. A posteriori, given an infinite sequence {R n } n 0 of local quadratic transforms of regular local rings, there exists a valuation ring V such that the sequence is along V. As usual, the 2-dimension case is far more tractable than the general case. If V is a valuation domain birationally dominating R 0, then the dimension inequality Equation 2.1 implies that tr. deg. R0 /m R0 V/m V is either 0 or 1; that is, either V is zero-dimensional over R 0 or V is a prime divisor of R 0. Assume by way of contradiction that S is not a valuation domain. By the result stated in the previous paragraph, there exists a valuation domain V birationally dominating S such that tr. deg. S/mS (V/m V ) > 0, so tr. deg. S/mS (V/m V ) = 1 and hence tr. deg. R0 /m 0 (V/m V ) = 1. This implies that V is divisorial, but then the sequence of local quadratic transforms along V must stabilize, so we have derived a contradiction. We have given Abhyankar s proof of [1, Lemma 12]: Corollary 4.3.3. [1, Lemma 12] Assume Setting 4.3.1. If dim R 0 = 2, then S is a valuation domain. However, if dim R 0 3, Shannon gives examples [27, Examples 4.7 and 4.17] that shows it is possible that S is not a valuation domain. This raises a natural question: under what conditions is S a valuation domain? The simplest case is if the sequence of blow-ups is along a DVR. Proposition 4.3.4. Assume Setting 4.3.1. If S is dominated by a DVR V (for instance, if the sequence is along a DVR), then S = V. This statement is well-known. We give the proof because it is simple and illustrative. Proof. Fix an element x V and for each n, write x = an b n in reduced form, where a n, b n R n. If ν(b n ) = 0, then b n is a unit in R n and it follows that x R n,

23 so it suffices to show ν(b n ) = 0 for some large n. Let n 0 such that ν(b n ) > 0 and write m n R n+1 = yr n+1. Since ν(a n ) ν(b n ) ν(y) > 0, it follows that a n /y R n+1, b n /y R n+1, and x = an/y b n/y. Therefore b n+1 divides bn y in R n+1, so ν(b n+1 ) ν(b n ) ν(m n ) ν(b n ) 1. Thus the sequence of nonnegative integers {ν(b n )} n 0 strictly decreases until it stabilizes at 0. The central idea in this proof is the inequality ν(b n+1 ) ν(b n ) ν(m n ). This same inequality holds if ν is an arbitrary rank 1 valuation, not necessarily a discrete one. Granja uses this idea to extend this result in [7, Proposition 23] for a rank 1 valuation V with the property that ν(m n ) =, with essentially the same proof. n=0 This observation provides a starting point for the tools we develop in Chapter 5. Proposition 4.3.5. [7, Proposition 23] Assume Setting 4.3.1. If S is dominated by a rank 1 valuation domain V with real valuation ν, and ν(m n ) =, then S = V. n=0 The question of whether S is a valuation domain depends on the behavior of height 1 primes of the rings R n. We use the following notation: Definition 4.3.6. Let R be an integrally closed Noetherian local domain. Then the essential prime divisors of R are epd(r) = {ν P P is a height 1 prime of R}, where ν P denotes the P -adic valuation, i.e. the discrete valuation associated to R P. Let ord Rn denote the order valuation of R n. Then ord Rn epd(r n+1 ). For m > n + 1, it is possible that either ord Rn epd(r m ) or ord Rn / epd(r m ). Notice epd(r n+1 ) epd(r n ) {ord Rn }. In this sense, the sequence {epd(r n )} n 0 is almost a descending sequence of sets. Consider the limit, Γ = epd(r m ) = (epd(r n ) {ord Rm m n}). (4.5) n 0 m n n 0

24 The number of elements of Γ determines whether or not S is a valuation ring, and if so, in addition determines the rank of S. Theorem 4.3.7. [27, Proposition 4.18], [6, Theorem 13, Proposition 14] Assume Setting 4.3.1 and let Γ be as in the preceding discussion. If Γ =, then S is a rank 1 valuation ring. If Γ = {ν P }, then S is a rank 2, rational rank 2 valuation ring and the valuation ring of ν P is its unique rank 1 valuation overring. Otherwise, Γ is infinite and S is not a valuation ring. If Γ =, then in Shannon s terminology, the sequence {R n } n 0 switches strongly infinitely often. If Γ = {ν P }, then in Granja s terminology, the sequence is height 1 directed along ν P. In the case where Γ is infinite, then at most n 1 of the valuation rings {ord Rn } n 0 are among Γ. This relates to the geometric notion of proximity. Definition 4.3.8. Let R be a regular local ring of dimension 2 and let S R be an infinitely near point. Then S is said to be proximate to R if S ord R, or equivalently, if ord R epd(s). The following proposition is well-known. See [16, Proposition 2.8] for an explicit purely algebraic proof. Proposition 4.3.9. For n > 0, the ring R n is proximate to at most dim R n of R 0, R 1,..., R n 1. It follows immediately that ord Rn Γ for at most finitely many Γ. Thus we conclude: Corollary 4.3.10. Γ epd(r n ) for n 0.

25 4.4 The Noetherian Hull The quadratic Shannon extension S is an integrally closed local domain that is generally not Noetherian. Despite not being Noetherian in general, it is locally Noetherian on its punctured spectrum SpecS \ {m S }. Let p SpecS \ {m S }. Since p = (p R n ) and p m S, it follows that p R n m n for n 0, and so n 0 (R n ) p Rn = (R n+1 ) p Rn+1 for n 0. Since S p = (R n ) p Rn, it follows that S p is a localization of R n for n 0, so it is an RLR. An element x m S is m S -primary that is, xs = m S if and only if x / p for all p SpecS \{m S }. The following lemma proves the existence of m S -primary elements. Lemma 4.4.1. Assume Setting 4.3.1. Let N 0 be such that Γ epd(r n ) for n N as in Corollary 4.3.10. For any n N, let x m n be such that m n R n+1 = xr n+1. Then xs = m S. Proof. Assume by way of contradiction that x p for some nonzero prime ideal p m S. Since S p is a Noetherian domain, Krull s Principal Ideal Theorem [23, Theorem 13.5] implies that there exists a height 1 prime q p such that x q. Since Γ epd(r n ), it follows that q R n is a height 1 prime. Then m n R n+1 = xr n+1 q R n+1, so q R n = (q R n+1 ) R n = xr n+1 R n = m n, and we have derived a contradiction. The existence of an m S -primary element shows that the punctured spectrum of S is the spectrum of the ring obtained by inverting such an m S -primary element. We give this ring a name: Definition 4.4.2. Let x m S be such that xs = m S. Then T = S[ 1 ] is called the x Noetherian hull of S. Remark 4.4.3. Assume Setting 4.3.1 and let T be the Noetherian hull of S. n 0 1. T is the unique minimal proper Noetherian overring of S. 2. For n 0, T is a localization of R n at a multiplicatively closed set.

26 3. T is a regular UFD. An element x m S generates an m S -primary ideal if and only if x T, i.e. {x m S xs = m S } = m S T. (4.6) 4.5 The Boundary Valuation We make extensive use of the following lemma, which shows, given a pair of elements in R 0, how to compare their orders in R n. Lemma 4.5.1. Assume Setting 4.3.1. Let I 0 = (a 0, b 0 )R 0 be an ideal of R 0, where a 0, b 0 R m are nonzero. For n 0, denote I n = (I 0 ) Rn such that am b m and write I n = (a n, b n )R n = an b n. Denote r = lim n ord Rn (I n ); r is a nonnegative integer. Then: 1. If ord Rn (a n ) ord Rn (b n ), then (a n R n ) R n+1 = a n+1 R n+1 and ord Rn (a n+1 ) ord Rn (a n ). 2. If ord Rk (a k ) = r for some k m, then ord Rn (a n ) = r for all k m. 3. Either ord Rn (a n ) = r for all n 0 or ord Rn (b n ) = r for all n 0. Proof. Item 1 follows readily from Equation 4.1 and Item 2 is an immediate consequence of Item 1. To see Item 3, notice that since ord Rn ((a n, b n )R n ) = r, either ord Rn (a n ) = r or ord Rn (b n ) = r. Now apply Item 2. Corollary 4.5.2. Under Setting 4.3.1, for nonzero a K, the sign of ord Rn (a) is constant for n 0. Proof. Write a = b c, where b, c R 0 have no common factors. The case b = 0 is trivial, so assume b 0. Then apply Lemma 4.5.1 to the ideal (b, c)r 0 to obtain ideals (b n, c n )R n, where bn c n = a for all n 0. Notice that for all n, we have ord Rn (a) = ord Rn (b n ) ord Rn (c n ).

27 Let r = lim n ord Rn ((b n, c n )R n ). By Lemma 4.5.1.3, there exists an N such that either ord Rn (b n ) = r, in which case ord Rn (a) 0 for all n N, or ord Rn (c n ) = r, in which case ord Rn (a) 0 for all n N. Lemma 4.5.1.2 implies that if ord Rk (a) = 0 for any k N, then ord Rn (a) = 0 for all n k. We conclude that the sign of ord Rn (a) is constant for n 0. Another way of phrasing Corollary 4.5.2 is that the sequence of order valuations {ord Rn } n 0 converges as in Definition 3.2.2. We record this as a theorem. Theorem 4.5.3. Assume Setting 4.3.1. The sequence of order valuations {ord Rn } n 0 converges in the patch topology. That is, V m = {a K ord Rn (a) 0 for n 0} n 0 m n is a valuation ring. Definition 4.5.4. We call the valuation ring of Theorem 4.5.3 the boundary valuation ring of S and denote it V. Remark 4.5.5. As every element in S eventually has nonnegative ord Rn -value, it follows that V dominates S. This implies that, though the sequence {R n } n 0 was defined to be an arbitrary sequence of local quadratic transforms, it is a posteriori the uniquely determined sequence of local quadratic transforms of R 0 along the boundary valuation ring V.

28 5. THE STRUCTURE OF A SHANNON EXTENSION In this chapter, we analyze the limits of the order valuation rings of Setting 4.3.1. We make the following assumptions and fix the following notation throughout this chapter: Setting 5.0.1. 1. {(R n, m n )} n 0 is an infinite sequence of local quadratic transforms of regular local rings of dimension d, where d 2. 2. S = R n denotes the directed union and m S = m n is its maximal ideal. n 0 3. V n S for all n, where V n is the order valuation ring of R n. (See Proposition 4.3.9.) 4. T denotes the Noetherian hull of the sequence as in Definition 4.4.2. 5. V denotes the boundary valuation ring of the sequence as in Definition 4.5.4. 6. Fix an m n -primary element x. We lose no generality in making the additional assumptions of Setting 5.0.1; we do so to simplify the statements of our results. Given an arbitrary sequence of local quadratic transforms of regular local rings, we may achieve these additional conditions by replacing R 0 with R n for some large value of n. Since dim R n = dim R n+1 for all n 0, it follows that R n+1 /m n+1 is a finite algebraic field extension of R n /m n. Thus S/m S is a (not necessarily finite) algebraic field extension of R 0 /m 0. The condition Setting 5.0.1.3 implies that the assumptions in Lemma 4.4.1 are met. We restate a simplified and extended form of Lemma 4.4.1 here, which relates the transforms of principal ideals and the set of m S -primary elements. n 0