IDEAL THEORY of LOCAL QUADRATIC TRANSFORMS of REGULAR LOCAL RINGS a Dissertation Submitted to the Faculty of Purdue University B

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IDEAL THEORY OF LOCAL QUADRATIC TRANSFORMS

OF REGULAR LOCAL RINGS

A Dissertation

Submitted to the Faculty

Purdue University

Matthew J. Toeniskoetter

In Partial Fulﬁllment of the

Requirements for the Degree

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 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

{Rn}n≥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

{ordRn }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 {ordRn }n≥0 with the sequence of Rees valuations {νn}n≥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 ∈ 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 {(Rn, mn)}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 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 {Rn}n≥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. 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. 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 {Rn}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

1 T = S[ x ] for any mS-primary element x ∈ mS. It is the minimal proper Noetherian overring of S.

Each of the rings Rn has an order valuation associated to its maximal ideal, denoted by ordRn . 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 aR0, we consider

Rn its sequence of successive transforms, {(aR0) }n≥0. The corresponding sequence of

Rn orders {ordRn ((aR0) )}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 mS-primary element x ∈ mS to set w(x) = 1. Then w is deﬁned by ord (a) w(a) = lim Rn . n→∞ ordRn (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 ∞ X τ = w(mn). 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 6= 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 = Sp. 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, mE).

2. w(−) induces a rank 1 rational rank 1 valuation w(−) that deﬁnes a valuation

ring W on the residue ﬁeld E/mE.

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 blow- up 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 {(Rn, mn)}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 ordR. By making

the assumption that the ideals mn have a unique Rees valuation νn, i.e. that they are one ﬁbered, we obtain a sequence {νn}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 {νn}n≥0 of Rees valuations converges.

Theorem 6.5.2. Let {(Rn, mn)}n≥0 be a sequence of local normalized quadratic transforms of analytically unramiﬁed Noetherian local domains. If mn has a unique 6

Rees valuation νn for each n ≥ 0, then the sequence {νn}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 mR 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, mR) and (S, mS) be local rings. A ring homomorphism φ : R → S is called −1 local if φ (mS) = mR. For an inclusion of local rings R ⊆ S, the ring S dominates

R if the inclusion map is local, i.e. if mS ∩ R = mR. 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, mR) be a Noetherian local domain and let (S, mS) be a local domain that dominates R. Then the following formula, called the dimension inequality [23, Theorem 15.5], holds:

dim S + tr. deg. (S/m ) ≤ dim R + tr. deg. Q(S). (2.1) R/mR S Q(R) 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 = P0 ( P1 ( ... ( Pn = 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. (S/m ) = dim R + tr. deg. Q(S). (2.2) R/mR S Q(R)

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] 6= [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 RΩ, where RΩ is the linearly ordered 9 abelian group of all real-valued functions from Ω to R that vanish outside of a well- ordered set.

A set S of elements of Γ is called rationally independent if for any Z-linear relation

p1x1 + ... + pnxn = 0 where x1, . . . , xn are distinct elements in S and p1, . . . , pn ∈ Z, then p1, . . . , pn = 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×) 6= {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 mV is the set of elements K with positive ν-value,

mV = {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 AmV ∩A, which implies that V has ﬁnite rank. By the Hahn embedding theorem, there exists an embedding Γ ,→ Rrank 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/mV/mV + r. rank ν ≤ dim R. (2.3)

Moreover, if Equation 2.3 is an equality, then the value group of ν is isomorphic as

dim R an (unordered) abelian group to Z and the residual ﬁeld extension R/m ⊆ V/mV 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 Zrank ν. A rank 1 discrete valuation is called a Discrete Valuation Ring (DVR) and has special impor- tance in commutative algebra and algebraic geometry. We recall equivalent conditions for a local ring to be a DVR: 11

Theorem 2.4.2. [23, Theorem 11.2] Let (R, m) be a local ring. The following are equivalent:

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

α1 αn ideal gcd(I) can be written in the form P1 ··· Pn , unique up to re-ordering, where

P1,...,Pn 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 AP contains B, in which case PAP ∩ B is a height 1 prime ideal and

AP = BPAP ∩B, or AP does not contain B. The transform of the height 1 prime ideal P in B is deﬁned, depending on these two cases, to be  PAP ∩ B if B ⊆ AP , P B = B otherwise.

For an ideal J ⊆ A of height ≥ 2, the transform of J in B is deﬁned to be

JB J B = . gcd(JB)

For general ideals I ⊆ A, the transform of I in B is deﬁned multiplicatively. Writing

I uniquely in the form I = P1 ··· PnJ, where htJ ≥ 2, the transform of I in B is deﬁned to be

B B B B I = P1 ··· Pn J . 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 IA = I.

2. If A ⊆ B is a birational inclusion of UFDs and I,J ⊆ A are ideals, then (IJ)B = IBJ B.

3. If A ⊆ B ⊆ C are birational inclusions of UFDs and I ⊆ A is an ideal, then (IB)C = IC .

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 =

α1 αn P1 ··· Pn J uniquely, where P1,...,Pn are distinct height 1 primes of A, α1, . . . , αn are nonzero (but possibly negative) integers, and J is an A-ideal of height ≥ 2. Then

B B α1 B αn B I = (P1 ) ··· (Pn ) J 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,

UA = {V ∈ V | A ⊂ V }.

These sets UA form an open basis for the Zariski topology. Notice that

−1 V\ U{x−1} = {V ∈ V | x ∈/ V } = {V ∈ V | x ∈ mV }.

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

UA,B = UA ∩ (V\ UB) = {V ∈ V | A ⊂ V,B ⊂ mV }.

These sets UA,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 {Vn}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 {Vn}n≥0 is convergent in a

topological sense, then it has a unique limit point lim Vn. Moreover, since the patch n→∞ 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 {Vn}n≥0 has at least one limit point, it is possible that no subsequence of

{Vn}n≥0 is convergent. We translate the topological condition of convergence to an algebraic one:

Deﬁnition 3.2.1. Let {Vn}n≥0 be a sequence of valuation rings. This sequence is convergent if and only if [ \ A = Vm n≥0 m≥n is a valuation ring. In this case, the sequence converges to the valuation ring A.

We restate the deﬁnition in terms of valuations:

Deﬁnition 3.2.2. Let {νn}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 νn(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 {Rn, mn}n≥0 be an inﬁnite birational dominating sequence of S Noetherian local domains (i.e. mn+1 ∩ Rn = mn) whose union Rn is a valuation n≥0 domain V . For each n ≥ 0, let Vn be a valuation ring that birationally dominates Rn.

Then the sequence of valuation rings {Vn}n≥0 converges to V . 15

Proof. For x ∈ V , it follows that x ∈ Rn for n 0, so x ∈ Vn for n 0. −1 −1 −1 For x∈ / V , since x ∈ V , it follows that x ∈ Rn for n 0, so x ∈ mn for −1 n 0. Since Vn dominates Rn, it follows that if x ∈ mVn for n 0, i.e. x∈ / Vn 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 mV = {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 x, y ∈ R such that ν(x) = ν(y) = ∞, the value of 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 {νn}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 {νn}n≥0 be a sequence of rank 1 valuations on K dominating R that converges to a valuation ring

V . Then for nonzero x ∈ mR, the limit

νn(y) ωx(y) = lim n→∞ νn(x)

exists for all y ∈ K, but may be ±∞ for nonzero values of y. The function ωx

is a possibly inﬁnite valuation whose valuation ring Vωx is a valuation overring of

V . Moreover, xVωx is mVωx -primary; in particular, Vωx = V if and only if xV is

mV -primary. 17

p Proof. Fix a nonzero y ∈ K. Let q ∈ Q be any rational number. Since the sequence yp νn converges, the sign of νn( xq ) is constant for n 0. Rewriting, the sign of pνn(y)− qν (x) is constant for n 0. That is, either νn(y) < p for n 0, νn(y) = p for n νn(x) q νn(x) q n 0, or νn(y) > p for n 0. Thus every rational number is an eventual upper νn(x) q or lower bound for the sequence { νn(y) } , so the sequence either converges to some νn(x) n≥0 real number or diverges to ±∞. Thus ωx(y) exists for all y ∈ K. To verify that ω is a p.i. valuation, let y, z ∈ K. The sequence { νn(yz) } equals x νn(x) n≥0 { νn(y) + νn(z) } , so ω (yz) = ω (y)+ω (z) unless this expression is ±(∞−∞). The νn(x) νn(x) n≥0 x x x sequence { νn(y+z) } is termwise greater than or equal to {min{ νn(y) , νn(z) }} , so νn(x) n≥0 νn(x) νn(x) n≥0

ωx(y + z) ≥ min{ωx(y), ωx(z)}. Thus ωx is a p.i. valuation.

To see that the valuation ring Vωx is a valuation overring of V , let y ∈ V . Then

νn(y) ≥ 0 for n 0 by deﬁnition of V . Since x ∈ mR and each of the νn dominate R, it follows that ν (x) > 0 for all n ≥ 0. Therefore νn(y) ≥ 0 for n 0, so ω (y) ≥ 0 n νn(x) x and y ∈ Vωx . Thus Vωx is a valuation overring of V .

Since x has positive ﬁnite value in Vωx , it is mVωx -primary. Since Vωx is a valuation √ overring of V , we conclude that Vωx = V xV , and in particular V = Vω 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 ωx maps to R ∪ {±∞}, the valuation ring corresponding to ωx 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[x1, . . . , xd] be a polynomial ring in d variables over a i ﬁeld k. For n ≥ 0, let νn be the real monomial valuation such that νn(xi) = n .

Then the sequence {νn}n≥0 converges to a rank d valuation ring described by the valuation ν : R → Zd, where Zd is ordered lexicographically. The valuation ν is

deﬁned by ν(xi) = (0,..., 1,..., 0), where the 1 is in the (d − i + 1)-th position, so

that νn(xd) = (1, 0,..., 0) has the largest value and νn(x1) = (0,..., 0, 1) has the

smallest value. The p.i. valuation ωxi 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

I selecting an element a ∈ I such that IV = aV , then localizing the ring R[ a ] at the 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 6⊂ p. If there exists a prime ideal q ∈ SpecS such that q ∩ R = p, then Rp = Sq. 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 (R0, m0) = (R, m). By iterating

local quadratic transforms, there exists a birational local sequence {(Rn, mn)}n≥0 of

Noetherian local domains, where (Rn+1, mn+1) is the local quadratic transform of 19

(Rn, mn) along V . Set (Rn+1, mn+1) to be the local quadratic transform of (Rn, mn) along V , so

R = R0 ⊆ R1 ⊆ · · · ⊆ Rn−1 ⊆ Rn ⊆ · · · .

If Rn+1 = Rn for any n ≥ 0, or equivalently, if mn is principal for any n ≥ 0, then

Rn 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 = Rn and the sequence stabilizes after a ﬁnite number of steps. If V 6= Rn for any n ≥ 0, then Rn ( Rn+1 for all n ≥ 0. Thus the sequence {(Rn, mn)}n≥0 of local quadratic transforms is an inﬁnite sequence of proper Noetherian overrings.

Assume that R = R0 is universally catenary, so the dimension formula Equa- tion 2.2 holds. Suppose that the sequence of local quadratic transforms along V stabilizes with Rn = 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 {(Rn, mn)}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 R0 ≥ 2. If (R0, m0) is an RLR and (R1, m1) is a local quadratic of m, then R1 is itself an RLR [1, Lemma 10]. Let {(Rn, mn)}n≥0 be the sequence of local quadratic transforms along a valuation V birationally dominating

R0. An RLR Rn of dimension ≥ 2 in that appears in any such sequence is an inﬁnitely near point of R0, denoted R0 Rn. 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 Rn = V , then V is the order valuation ring of Rn−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 R0 be a local quadratic transform of R. Since the rings R and R0 are UFDs, we have the notion of the transform of an R-ideal or a fractional R-ideal from R to R0 as in Section 2.2.5. Let J ⊆ K be such a fractional ideal. Then the transform J R0 of the fractional R-ideal J in the ring R0 is the ideal given the by equation

0 mordR(J)J R = JR0. (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 mR0 is principal. By ﬁxing an element x ∈ m such that mR0 = xR0, we may rewrite this equation as

0 0 JR J R = . (4.2) xordR(J) 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 R0 be a local quadratic

R0 transform of R. If J ⊆ R is an R-ideal, then ordR0 (J ) ≤ ordR(J).

For an iterated sequence of local quadratic transforms, we can iterate the process

m of taking the transform of an ideal. Let {(Rn, mn)}n=0 be a ﬁnite sequence of local

quadratic transforms and let J be a fractional R0-ideal. Then

"m−1 # Rn Y ordRn (J ) Rm mn J = JRm. (4.3) n=0 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 xn ∈ mn such that mnRm = xnRm for 0 ≤ n < m, we may rewrite this equation as JR J Rm = m . (4.4) m−1 Rn Q ordRn (J ) xn n=0 21

Given an inﬁnite sequence {(Rn, mn)}n≥0 of local quadratic transforms, and an

Rn R0-ideal J ⊆ R0, then the sequence {ordRn (J )}n≥0 is a non-increasing sequence

of nonnegative integers, so it eventually stabilizes. Since any fractional R0-ideal −1 0 J ⊆ K can be written in the form J = a J for some a ∈ R0 and some R0-ideal

J ⊆ R0, and as transforms of ideals are multiplicative, it follows that the sequence

Rn {ordRn (J )}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 {(Rn, mn)}n≥0 be an inﬁnite sequence of local quadratic transforms of regular local rings of dimension ≥ 2,

R0 ( R1 ( ··· ( Rn ( ··· .

S S Denote S = Rn and mS = mn. 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 mS. 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 xn ∈ mn such that mnRn+1 = xnRn+1, and notice that

xn−1Rn+1 ⊆ mnRn+1 = xnRn+1. Hence x0S ⊆ x1S ⊆ ... is an inﬁnite ascending chain of ideals in S. If S is Noetherian, then this chain stabilizes, so there exists k S such that xkS = (x1, x2,...)S. But mS = mn = (x1, x2,...)S = xkS, so mS is n≥0 principal. Since S is a Noetherian local domain whose maximal ideal is principal, S is a DVR [23, Theorem 11.2]. 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. (V/m ) > 0 [29, Ch. IV §5 Theorem 10]. A posteri- S/mS V ori, given an inﬁnite sequence {Rn}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 R0, then the dimension inequality

Equation 2.1 implies that tr. deg. V/mV is either 0 or 1; that is, either V is R0/mR0

zero-dimensional over R0 or V is a prime divisor of R0. 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. (V/m ) > 0, so tr. deg. (V/m ) = 1 and hence tr. deg. (V/m ) = S/mS V S/mS V R0/m0 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 R0 = 2, then S is a valuation domain.

However, if dim R0 ≥ 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 illustra- tive.

Proof. Fix an element x ∈ V and for each n, write x = an in reduced form, where bn

an, bn ∈ Rn. If ν(bn) = 0, then bn is a unit in Rn and it follows that x ∈ Rn, 23

so it suﬃces to show ν(bn) = 0 for some large n. Let n ≥ 0 such that ν(bn) >

0 and write mnRn+1 = yRn+1. Since ν(an) ≥ ν(bn) ≥ ν(y) > 0, it follows that a /y ∈ R , b /y ∈ R , and x = an/y . Therefore b divides bn in R , so n n+1 n n+1 bn/y n+1 y n+1

ν(bn+1) ≤ ν(bn) − ν(mn) ≤ ν(bn) − 1. Thus the sequence of nonnegative integers

{ν(bn)}n≥0 strictly decreases until it stabilizes at 0.

The central idea in this proof is the inequality ν(bn+1) ≤ ν(bn) − ν(mn). 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 ∞ P valuation V with the property that ν(mn) = ∞, 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

∞ X ν(mn) = ∞, n=0 then S = V .

The question of whether S is a valuation domain depends on the behavior of height

1 primes of the rings Rn. 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) = {νP | P is a height 1 prime of R}, where νP denotes the P -adic valuation, i.e. the discrete valuation associated to RP .

Let ordRn denote the order valuation of Rn. Then ordRn ∈ epd(Rn+1). For m > n + 1, it is possible that either ordRn ∈ epd(Rm) or ordRn ∈/ epd(Rm).

Notice epd(Rn+1) ( epd(Rn)∪{ordRn }. In this sense, the sequence {epd(Rn)}n≥0 is ”almost” a descending sequence of sets. Consider the limit,

[ \ \ Γ = epd(Rm) = (epd(Rn) ∪ {ordRm | 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 = {νP }, then S is a rank 2, rational rank 2 valuation ring and the valuation

ring of νP 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 {Rn}n≥0 switches strongly inﬁnitely often. If Γ = {νP }, then in Granja’s terminology, the sequence is height 1 directed along νP .

In the case where Γ is inﬁnite, then at most n−1 of the valuation rings {ordRn }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 ⊆ ordR, or

equivalently, if ordR ∈ 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 Rn is proximate to at most dim Rn of

R0,R1,...,Rn−1.

It follows immediately that ordRn ∈ Γ for at most ﬁnitely many Γ. Thus we conclude:

Corollary 4.3.10. Γ ⊂ epd(Rn) 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 \{mS}. Let p ∈ SpecS \{mS}. Since S p = (p ∩ Rn) and p 6= mS, it follows that p ∩ Rn ( mn for n 0, and so n≥0 S (Rn)p∩Rn = (Rn+1)p∩Rn+1 for n 0. Since Sp = (Rn)p∩Rn , it follows that Sp is a n≥0 localization of Rn for n 0, so it is an RLR. √ An element x ∈ mS is mS-primary–that is, xS = mS–if and only if x∈ / p for all p ∈ SpecS \{mS}. The following lemma proves the existence of mS-primary elements.

Lemma 4.4.1. Assume Setting 4.3.1. Let N ≥ 0 be such that Γ ⊆ epd(Rn) for n ≥ N

as in Corollary 4.3.10. For any n ≥ N, let x ∈ mn be such that mnRn+1 = xRn+1. √ Then xS = mS.

Proof. Assume by way of contradiction that x ∈ p for some nonzero prime ideal p ( mS. Since Sp 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(Rn), it follows that q ∩ Rn is a height 1 prime. Then mnRn+1 = xRn+1 ⊆

q ∩ Rn+1, so q ∩ Rn = (q ∩ Rn+1) ∩ Rn = xRn+1 ∩ Rn = mn, and we have derived a contradiction.

The existence of an mS-primary element shows that the punctured spectrum of S

is the spectrum of the ring obtained by inverting such an mS-primary element. We give this ring a name: √ 1 Deﬁnition 4.4.2. Let x ∈ mS be such that xS = mS. Then T = S[ x ] is called the Noetherian hull of S.

Remark 4.4.3. Assume Setting 4.3.1 and let T be the Noetherian hull of S.

1. T is the unique minimal proper Noetherian overring of S.

2. For n 0, T is a localization of Rn at a multiplicatively closed set. 26

3. T is a regular UFD.

× An element x ∈ mS generates an mS-primary ideal if and only if x ∈ T , i.e. √ × {x ∈ mS | xS = mS} = mS ∩ T . (4.6)

4.5 The Boundary Valuation

We make extensive use of the following lemma, which shows, given a pair of

elements in R0, how to compare their orders in Rn.

Lemma 4.5.1. Assume Setting 4.3.1. Let I0 = (a0, b0)R0 be an ideal of R0, where

Rn a0, b0 ∈ Rm are nonzero. For n ≥ 0, denote In = (I0) and write In = (an, bn)Rn

am an such that = . Denote r = lim ordR (In); r is a nonnegative integer. Then: bm bn n→∞ n

Rn+1 1. If ordRn (an) ≤ ordRn (bn), then (anRn) = an+1Rn+1 and ordRn (an+1) ≤

ordRn (an).

2. If ordRk (ak) = r for some k ≥ m, then ordRn (an) = r for all k ≥ m.

3. Either ordRn (an) = r for all n 0 or ordRn (bn) = 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 ordRn ((an, bn)Rn) = r, either ordRn (an) = r or

ordRn (bn) = r. Now apply Item 2.

Corollary 4.5.2. Under Setting 4.3.1, for nonzero a ∈ K, the sign of ordRn (a) is constant for n 0.

b Proof. Write a = c , where b, c ∈ R0 have no common factors. The case b = 0 is trivial, so assume b 6= 0. Then apply Lemma 4.5.1 to the ideal (b, c)R0 to obtain ideals (b , c )R , where bn = a for all n ≥ 0. Notice that for all n, we have ord (a) = n n n cn Rn

ordRn (bn) − ordRn (cn). 27

Let r = lim ordR ((bn, cn)Rn). By Lemma 4.5.1.3, there exists an N such that n→∞ n either ordRn (bn) = r, in which case ordRn (a) ≤ 0 for all n ≥ N, or ordRn (cn) = r, in which case ordRn (a) ≥ 0 for all n ≥ N. Lemma 4.5.1.2 implies that if ordRk (a) = 0 for any k ≥ N, then ordRn (a) = 0 for all n ≥ k. We conclude that the sign of ordRn (a) is constant for n 0.

Another way of phrasing Corollary 4.5.2 is that the sequence of order valuations

{ordRn }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 {ordRn }n≥0 converges in the patch topology. That is,

[ \ Vm = {a ∈ K | ordRn (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 ordRn -value, it follows that V dominates S. This implies that, though the sequence {Rn}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 R0 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. {(Rn, mn)}n≥0 is an inﬁnite sequence of local quadratic transforms of regular local rings of dimension d, where d ≥ 2.

S S 2. S = Rn denotes the directed union and mS = mn is its maximal ideal. n≥0 n≥0

3. Vn 6⊃ S for all n, where Vn is the order valuation ring of Rn. (See Proposi- tion 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 mn-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 R0 with Rn for some large value of n.

Since dim Rn = dim Rn+1 for all n ≥ 0, it follows that Rn+1/mn+1 is a ﬁnite

algebraic ﬁeld extension of Rn/mn. Thus S/mS is a (not necessarily ﬁnite) algebraic

ﬁeld extension of R0/m0. 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 mS-primary elements. 29

Proposition 5.0.2. Assume the notation of Setting 5.0.1. Then

1. mnS is mS-primary for each n ≥ 0.

Rn 2. If y ∈ mm, then yS is mS-primary if and only if (yRm) = Rn for n 0.

× Rn 3. If y ∈ K, then y ∈ T if and only if (yRm) = Rn for n 0.

Rn Rn 4. If y ∈ K and m ≥ 0, then (yR0) = (yRm) for n 0.

Proof. Since the conditions of Lemma 4.4.1 are met, the conclusion Proposition 5.0.2.1 follows. n−1 Rn Q αi For Item 2, since yRn = (yRm) mi for some non-negative integers i=m αm, . . . , αn−1, the ”if” direction holds by Item 1. To see the ”only if” direction,

assume that yS is mS-primary and consider the sequence of sets {Γn}n≥m, where Γn

Rn denotes the finite set of DVRs associated to the height 1 prime ideals Min((yRm) ). T Then Γn is a subset of Γ, where Γ is defined as in Equation 4.5. Since yS is n≥0 T mS-primary, y is not contained in any of the DVRs of Γ, so Γn = ∅. Since the Γn n≥m are a descending sequence of sets by definition of transform, we conclude that Γn = ∅

Rn for n 0, i.e. that (yRm) = Rn for n 0. Item 3 is equivalent to Item 2 as in Equation 4.6. Item 4 follows from Item 1, Item 3, and Equation 4.3.

5.1 The Intersection Decomposition

In Deﬁnition 4.4.2 and Deﬁnition 4.5.4, we introduced the Noetherian hull and the boundary valuation of the sequence in Setting 5.0.1, respectively. We show the utility of these constructions by proving that S is the intersection of these two rings.

Theorem 5.1.1. Assume the notation of Setting 5.0.1. Then

S = V ∩ T.

Proof. It is clear that S ⊆ T , and S ⊆ V by Remark 4.5.5, so S ⊆ T ∩ V . 30

1 b To see S ⊇ V ∩ T , let a ∈ V ∩ T . Since a ∈ T and T = S[ x ], we may write a = xk

for some b ∈ S and some integer k ≥ 0. Let m ≥ 0 such that b, x ∈ Rm and consider

k Rn the ideal I = (b, x )Rm. Write In = I = (bn, cn) as in Lemma 4.5.1 for all n ≥ m,

so ordRn (a) = ordRn (bn) − ordRn (cn). Since ordRn (a) ≥ 0 for n 0 by deﬁnition

of V , it follows that ordR (bn) ≥ ordR (cn) for n 0. Let r = lim ordR (In), so n n n→∞ n

Lemma 4.5.1.3 implies that ordRn (cn) = r for n 0.

cn For any m ≥ n, since cn+1 = r for some nonnegative integer rn ≥ r and some xn xn

with mnRn+1 = xnRn+1, and since xn is mS-primary by Proposition 5.0.2.1, it follows × inductively that cn ∈ T for all n ≥ m.

Let N be suﬃciently large such that ordRN (cN ) = r. Proposition 5.0.2.3 implies

Rn that ordRn ((cN RN ) ) = 0 for n 0, so Lemma 4.5.1.1 implies that r = 0.

Since r = 0, it follows that ordRn (cn) = 0 for n 0, i.e. cn is a unit in Rn for n 0. Thus a = bn ∈ R for n 0, so a ∈ S. cn n

5.2 The Asymptotic Limit of Order Valuations

We have seen in Theorem 4.5.3 that the sequence {ordRn }n≥0 of order valuations converges in the patch topology as in Definition 3.2.2. Then we may apply Theo- rem 3.3.4 to this sequence and to the fixed mS-primary element x ∈ S to obtain a possibly infinite valuation that partially describes V .

Theorem 5.2.1. Assume notation as in Setting 5.0.1. Then for a ∈ K, the limit ord (a) w(a) = lim Rn n→∞ ordRn (x) exists, but may be ±∞ for a 6= 0.

Proof. Theorem 4.5.3 and Theorem 3.3.4.

Deﬁnition 5.2.2. We denote by w(−) the possibly inﬁnite valuation of Theorem 5.2.1 and by W the valuation ring induced by w.

As in Theorem 3.3.4, W is a valuation overring of V , x is mW -primary, and W = V

if and only if x is mV -primary. 31

The following proposition shows that the possibly inﬁnite valuation w is always ﬁnite on the units of T .

Proposition 5.2.3. Assume notation as in Setting 5.0.1 and let w be as in Deﬁni- tion 5.2.2. Then

1. If a ∈ S, then w(a) ≥ 0, and if a ∈ mS, then w(a) > 0; that is, W birationally dominates S.

2. If a ∈ mS and a is mS-primary, then 0 < w(a) < ∞.

3. If a ∈ T ×, then w(a) is ﬁnite.

4. If a ∈ T ×, then a ∈ S if and only if w(a) ≥ 0.

Proof. For the ﬁrst item, since V ⊆ W , it is clear that S ⊆ W and that if a ∈ S, then

w(a) ≥ 0. To prove the remaining part, let a ∈ mS. Since x is mS-primary, there an an 1 exists n > 0 such that x ∈ mS. Thus w( x ) = nw(a) − 1 ≥ 0, so w(a) ≥ n , so the ﬁrst item holds.

For the second item, let a ∈ mS be mS-primary. Then, similarly to the ﬁrst item, xm xm xn there exists m > 0 such that a ∈ mS. Hence w( a ) ≥ 0, so w( a ) = n − w(a) ≥ 0. It follows that w(a) ≤ n < ∞. The third item follows from the multiplicativity of w and the fact that units of T

are precisely the ratios of mS-primary elements. To see the fourth item, let a ∈ T ×. The ﬁrst item implies that ”only if” direction. y For the ”if” direction, assume that w(a) ≥ 0. We may write a = z , where y, z ∈ mS are mS-primary. We proceed as in the proof of Theorem 5.1.1. Let m be suﬃciently large such that y, z ∈ Rm, then consider the transforms In = (yn, zn) of the ideal I =

(y, z)Rm in the rings Rn for n ≥ m. As in the proof of Theorem 5.1.1, it follows that × yn, zn ∈ T for n 0 and ordRn (In) = 0 for n 0. Assume by way of contradiction

that ordRn (zn) > 0 for all n ≥ 0, so since ordRn (In) = 0 for n 0, it follows

that ordRn (yn) = 0 for n 0. Fix N such that ordRN (yN ) = 0 and zN ∈ mN . Since × zN ∈ T ∩mS, it follows that zN is mS-primary as in Equation 4.6, so the second item 32

y implies that w(zN ) > 0. But yN ∈ mS, so w(yN ) = 0, so w( z ) = w(yN ) − w(zN ) < 0, contradicting the original assumption. Thus ord (z ) = 0 for n 0, so 1 ∈ R for Rn n zn n n 0, so y = yn ∈ R for n 0. z zn n

Remark 5.2.4. The set of principal mS-primary ideals are naturally ordered by inclusion, and this order extends to the natural order of principal fractional ideals of S generated by units of T . Indeed, this order makes the factor group T ×/S× into a rank 1 linearly ordered abelian group. Proposition 5.2.3.4 shows that the natural order on principal mS-primary ideals is precisely determined by w. This is analogous to the relationship between valuation rings and valuations. If V is a valuation ring on a ﬁeld K, then inclusion of principal fractional ideals of V induces an ordered abelian group structure on the factor group K×/V ×, and the natural projection ν : K× → K×/V × is a valuation whose induced valuation ring is V .

The w-value combined with the divisibility of elements in T completely determines the inclusions of principal ideals in S.

Proposition 5.2.5. Assume notation as in Setting 5.0.1. Let y, z ∈ S.

z z 1. yS ( zS (i.e. y ∈ mS) if and only if w( y ) > 0 and yT ⊆ zT .

z × z 2. yS = zS (i.e. y ∈ S ) if and only if w( y ) = 0 and yT = zT .

Proof.

z (1, ⇒) Assume that yS ( zS. Clearly yT ⊆ zT . Since y ∈ mS, Proposition 5.2.3.1 z implies that w( y ) > 0. z z z (1, ⇐) Assume that w( y ) > 0 and yT ⊆ zT . Thus y ∈ T . Since w( y ) > 0, z z y ∈ mV . It follows that y ∈ mS by Theorem 5.1.1 and Remark 4.5.5. z × (2, ⇒) Assume that yS = zS, so that y ∈ S . It is clear that yT = zT , and since z W contains S, it is also clear that w( y ) = 0. z z × (2, ⇐) Assume that w( y ) = 0 and yT = zT , so that y ∈ T . Then Proposi- z × tion 5.2.3.4 implies that y ∈ S . 33

5.3 The Limit of Transforms

Assume notation as in Setting 5.0.1 and let a ∈ K be nonzero. We have previously

Rn seen in Section 4.2 that the sequence of integers {ordRn (aR0) }n≥0 eventually stabilizes. We denote e(a) to be this limit.

Deﬁnition 5.3.1. For nonzero a ∈ K, deﬁne

Rn e(a) = lim ordR ((aR0) ). n→∞ n

Notice that Proposition 5.0.2.4 implies the starting point does not matter. That is, for any m ≥ 0,

Rn e(a) = lim ordR ((aRm) ). n→∞ n Since both orders and transforms are multiplicative, the function e(−) is multiplicative; for nonzero a, b ∈ K, it follows that e(ab) = e(a) + e(b).

Rn Notice that if a ∈ S, i.e. a ∈ Rm for some large m, then ordRn ((aRm) ) ≥ 0 for all n ≥ m, showing that e(a) ≥ 0. Moreover, for a ∈ mS, Proposition 5.0.2.2 shows that a is mS-primary if and only if e(a) = 0. Shannon’s condition for S to be a rank 1 valuation ring can be restated in terms of the function e(−): the ring S is a rank 1 valuation ring if and only if e(a) = 0 for all nonzero a ∈ K. Notice that if dim S ≥ 2, then there exists a ∈ S with e(a) > 0. Like the order valuation of a regular local ring, e(−) behaves like a degree function

a on the divisors of principal ideals. Let f ∈ T be nonzero and write f = b , where × a ∈ S and b ∈ T , so e(f) = e(a). Fix m such that a ∈ Rm and write aRm =

P1 ··· PrQ1 ··· Qs, where P1,...,Pr are height 1 primes of Rm such that PiT 6= T

and Q1,...,Qs are height 1 primes of Rm such that QiT = T . The divisor of f r P in T is the formal sum PiT . Since e(Qi) = 0 for 1 ≤ i ≤ s, it follows that i=1 r P e(f) = e(a) = e(Pi). In the special case where T is local with maximal ideal mT i=1 and mT ∩ Rn is a regular prime ideal for n 0, we go on to prove in Theorem 5.7.2 that e(−) is in fact the order valuation of T . 34

We make a simple remark that relates the e-value and the V -value of transforms. If the e-value of a is zero, then its transform eventually has zero V -value, and if the e-value of a is positive, then its transform eventually has positive V -value.

Remark 5.3.2. Assume notation as in Setting 5.0.1. Let a ∈ K be nonzero.

Rn 1. If e(a) = 0, then for n 0, for all m ≥ n, it follows that ordRm ((aR0) ) = 0;

Rn i.e. (aR0) V = V for m 0.

Rn 2. If e(a) > 0, then for n 0, for all m ≥ n, it follows that ordRm ((aR0) ) > 0;

Rn i.e. (aR0) V ⊆ mV for m 0.

Rm Proof. In either case, let n be large enough such that ordRm ((aR0) ) = e(a) for all m ≥ n. Then Equation 4.3 implies that, for all m ≥ n,

m−1 ! Rn Y e(a) Rm (aR0) = mi (aR0) . i=n

Rn Rm If e(a) = 0, then ordRm ((aR0) ) = ordRm ((aR0) ) = 0 for all m ≥ n. If e(a) > 0,

Rn then since ordRm (mi) > 0 for all m ≥ i ≥ 0, it follows that ordRm ((aR0) ) > 0 for all m ≥ n.

5.4 The Transform Formula for w

Let a ∈ K be nonzero and consider the fractional R0-ideal aR0. Then there is an associated sequence of orders,

Rn {rn := ordRn ((aR0) )}n≥0, (5.1)

where the rn are integers that stabilize to e(a). Recall that by Equation 4.3,

n−1 ! Y ri Rn aRn = mi (aR0) , i=0 hence n−1 X Rn w(a) = riw(mi) + w((aR0) ). i=0 35

By considering the limit as n goes to inﬁnity, we prove in the following theorem that

w(a) is completely determined by the sequence {rn}n≥0.

Theorem 5.4.1. Assume notation as in Setting 5.0.1. Let a ∈ K and let {rn}n≥0 be as in the preceding discussion. Then

∞ X w(a) = rnw(mn). n=0

Proof. We ﬁrst consider the case where e(a) = 0. Since rn = 0 for n 0, the series

Rn in the statement of the theorem is a ﬁnite sum, and by Remark 5.3.2, w((aR0) ) = 0 for n 0. We conclude that the theorem holds in this case. It remains to show the theorem where e(a) 6= 0. By possibly replacing a with

1 a , we assume that e(a) > 0. We break the theorem up into two remaining cases, ∞ P depending on whether w(mn) is finite or infinite. n=0 ∞ P In the first case, assume that w(mn) is finite. This implies that lim w(mn) = 0. n=0 n→∞ We first consider the theorem under the assumption a ∈ R0. Since e(a) > 0, it follows

Rn from Proposition 5.2.3.1 that w((aR0) ) > 0 for all n ≥ 0. Thus the ”≥” inequality of the statement follows. To see the ”≤” inequality, it suﬃces to show that for

Ri > 0, it follows that w((aR0) ) < for i 0. Let > 0. Since w(a) > 0, we

may assume < w(a). Since lim w(mn) = 0, there exists an mS-primary element n→∞ a y such that w(a) > w(y) > w(a) − . Now consider y . By Proposition 5.2.5.1, the a a element y is in S, so y ∈ Rn for some large n. Since taking transforms factors out

a Ri a elements of positive w-value, we have that w(( y Rn) ) < w( y ) < for all i ≥ n.

Ri Ri We have that (aR0) = (aRn) for i 0 by Proposition 5.0.2.4, and we have

Ri Ri a Ri a Ri that (aRn) = (yRn) ( y Rn) = ( y Rn) for i 0 by Proposition 5.0.2.2. Thus

Ri w((aRm) ) < for i 0, completing the proof in this case. ∞ P b In the case where a ∈ K and w(mi) < ∞, we may write a = c for some i=0 b, c ∈ R0, and the theorem follows by the multiplicativity of w and of the terms ri. 36

∞ P For the remaining case, let a ∈ K and assume that w(mi) = ∞. For any n ≥ 0, i=0 we have n X Ri Rn w(a) = ordRi (aR0) w(mi) + w((aR0) ). i=0

Ri Since ordRi ((aR0) ) stabilizes to e(a) > 0, the sum in the statement of the theorem

Rn diverges to ∞. By Remark 5.3.2, it follows that w((aR0) ) ≥ 0 for n 0, so the inﬁnite sum is a lower bound and the claim holds. This completes the proof.

As a direct corollary to Theorem 5.4.1, for an element a ∈ K, if e(a) = 0, then w(a) is always finite, but if e(a) 6= 0, then the finiteness of w(a) is determined by the ∞ P finiteness of w(mn). We give this constant a name. n=0 ∞ P Definition 5.4.2. Denote τ = w(mn). n=0 The following is an immediate corollary of Theorem 5.4.1.

Corollary 5.4.3. Assume notation as in Setting 5.0.1. If a ∈ K is nonzero, then there exists u ∈ T × such that

w(a) = w(u) + τe(a).

In particular, if τ < ∞, then w(a) is always ﬁnite, and if τ = ∞, then w(a) is ﬁnite if and only if e(a) = 0.

If w has rational rank ≥ 2, we can give an explicit upper bound for τ.

Proposition 5.4.4. [19, Proposition 5.12] Assume notation as in Setting 5.0.1. Let

y1, . . . , yr ∈ R0 and assume that w(y1), . . . , w(yr) are ﬁnite and rationally independent. Then r P w(xi) τ ≤ i=1 . r − 1 Proof. We argue as in the proof of [15, Prop. 7.3]. We inductively prove that for (n) (n) (n) (n) all n ≥ 0, there are elements y1 , . . . , yr ∈ mn such that w(y1 ), . . . , w(yr ) are rationally independent and n−1 ! r r X X (n) X (r − 1) w(mi) + w(yj ) ≤ w(yj). (5.2) i=0 j=1 j=1 37

(0) Taking yj = yj, the base case n = 0 is clear. Assume the claim is true for n. Thus (n) we have elements yj ∈ mn such that Equation 5.2 holds.

Let xn ∈ mn be such that xnRn+1 = mnRn+1. It then follows that the set (n) (n) {w(xn), w(y1 ), . . . , w(yr )} has rational rank at least r. By re-ordering, we may (n) (n) assume without loss of generality that w(xn), w(y2 ), . . . , w(yr ) are rationally in- y(n) dependent. Set y(n+1) = x and set y(n+1) = j for 2 ≤ j ≤ n. It follows 1 n j xn (n+1) (n+1) (n) that w(y1 ), . . . , w(yr ) are rationally independent. Since w(xn) < w(yj ) for (n+1) (n) 2 ≤ j ≤ n, it follows that yj ∈ mn+1 for 2 ≤ j ≤ n. Since y1 ∈ mn, it also follows (n) that w(xn) ≤ w(y1 ). Putting this all together, n ! r n−1 ! X X (n+1) X (r − 1) w(mi) + w(yj ) = (r − 1) w(mi) + (r − 1)w(mn)+ i=0 j=1 i=0 r X (n) w(mn) + (w(yj ) − w(mn)) j=2 n−1 ! r X X (n) ≤ (r − 1) w(mi) + w(yj ) i=0 j=1 r X ≤ w(yj), j=1 where the last inequality is a consequence of Equation 5.2. The conclusion follows.

5.5 The Archimedean Case

In Theorem 5.5.2, we construct an explicit valuation for the boundary valuation V in the case where τ < ∞. The valuation is simply (w(−), −e(−)), ordered lexicographically. By applying Theorem 5.4.1, using the assumption that τ < ∞, we prove that an element a ∈ Rm with ordRm (a) = e(a) has the unique minimal possible V -value among all elements in Rm with the same w-value. This is the crucial step toward proving Theorem 5.5.2.

Lemma 5.5.1. Assume notation as in Setting 5.0.1. Let m ≥ 0 and let a ∈ Rm be

such that ordRm (a) = e(a). If y ∈ Rm is such that w(y) = w(a), then either 38

1. e(y) < e(a), and ordRn (y) > ordRn (a) for all n ≥ m, or

2. e(y) = e(a), and ordRn (y) = ordRn (a) for all n ≥ m.

Proof. We have by Theorem 5.4.1 that

∞ X Rn w(a) = ordRn ((aRm) )w(mn) n=m ∞ X = e(a)w(mn), n=m ∞ X Rn w(y) = ordRn ((yRm) )w(mn). n=m

Rn Write rn = ordRn ((yRm) ). We use the fact that {rn}n≥m is a non-increasing sequence of nonnegative integers that stabilizes to e(y).

Certainly, since w(y) = w(a), it follows that ordRm (y) ≥ ordRm (a) and e(y) ≤ e(a). Since the inﬁnite sums are equal, either e(y) = e(a), which implies that each term is

equal and hence ordRm (y) = ordRm (a), or e(y) < e(a), which implies that the ﬁrst

term for y is strictly greater than e(a), i.e. ordRm (y) > ordRm (a). It remains to show

the inequality and equality for ordRn for n ≥ m. Write I = (a , y )R = ((a, y)R )Rn for n ≥ m as in Lemma 4.5.1, so a = an for n n n n m y yn

all n ≥ m. We have ordRn (a) − ordRn (y) = ordRn (an) − ordRn (yn) for each n ≥ m,

so to complete the proof, it suﬃces to show the inequality and equality for an and yn for all n ≥ m.

In the case where ordRm (y) = ordRm (a) = e(y) = e(a), Lemma 4.5.1.1 implies

that ordRn (an) = ordRn (yn) for all n ≥ m by a simple induction.

In the other case, suppose e(y) < e(a), or equivalently, ordRm (y) > ordRm (a).

By Lemma 4.5.1.1, it follows that ordRm+1 (am+1) = e(a). As w(am+1) = w(ym+1), it

follows that am+1 and ym+1 satisfy the same conditions as a and y, and since e(y) <

e(a), it follows that ordRm+1 (ym+1) > ordRm+1 (am+1). By induction, we conclude that

ordRm (y) > ordRm (a) for all m ≥ n. 39

Theorem 5.5.2. Assume notation as in Setting 5.0.1 and assume that τ < ∞. Then the following is a valuation for V :

× v : F → R ⊕ Z f 7→ (w(f), −e(f))

Proof. Since τ < ∞, Corollary 5.4.3 implies that w(a) < ∞ for all nonzero a ∈ K, proving that w is a rank 1 valuation. As w determines a valuation overring of V , it follows that w determines the rank 1 valuation overring of V . To prove that ν is a valuation for V , it suﬃces to consider elements a ∈ V with w(a) = 0. We shall

show that if a ∈ mV , then e(a) < 0, and if a ∈ V \ mV , then e(a) = 0, and this will complete the proof.

b Write a = c , where b, c ∈ R0, and for n ≥ 0, consider the transform In = (b , c )R = ((b, c)R )Rn as in Lemma 4.5.1. In particular a = bn and w(b ) = w(c ). n n n 0 cn n n

In the ﬁrst case, suppose that a ∈ mV , so ordRn (cn) < ordRn (bn) for n 0.

By Lemma 4.5.1.1, ordRn (cn) = e(cn) for n 0. Then Lemma 5.5.1 implies that e(b) < e(c), i.e. e(a) < 0.

In the second case, suppose that a ∈ V \ mV , so ordRn (bn) = ordRn (cn) for n 0.

Similarly, Lemma 4.5.1.1 implies that ordRn (bn) = e(bn) and ordRn (cn) = e(cn) for n 0. Thus e(b) = e(c), i.e. e(a) = 0.

Theorem 5.5.2 implies that if τ < ∞, then V has either rank 1 or rank 2. If e(a) = 0 for all nonzero a ∈ K, i.e. if S = V is a rank 1 valuation ring, then it is

clear that V has rank 1. However, it is possible for V to have rank 1 even if S ( V , if every f ∈ F such that w(f) = 0 also has that e(f) = 0. By Corollary 5.4.3, we may write w(f) = w(u) + τe(f) for some u ∈ T ×. Thus the existence of elements with w(f) = 0 and e(f) 6= 0 depends on whether τ is rationally dependent on w(T ×). We examine this in Section 5.6 and show in Example 5.8.3 that it is possible that V has rank 1 even if S ( V . 40

The condition that τ < ∞ is equivalent to the condition that S is an archimedean domain. We give the deﬁnition of an archimedean domain and some background about archimedean domains.

Deﬁnition 5.5.3. A domain A is called archimedean if for every nonzero nonunit a ∈ A, the equality T anA = (0) holds. n≥0 All Noetherian domains are archimedean by Krull’s Intersection Theorem [23, Theorem 8.9]. In addition, 1-dimensional domains, even if non-Noetherian, are archimedean, so non-archimedean domains must be non-Noetherian domains of dimension at least 2. Valuation domains of rank 2 or greater are examples of non- archimedean domains. In Setting 5.0.1, the invariant τ determines whether or not S is archimedean.

Theorem 5.5.4. Assume notation as in Setting 5.0.1 and assume that dim S ≥ 2. Denote p = {y ∈ S | w(y) = ∞}. Then √ \ \ n p = xS, and p = x S for any x ∈ mS such that xS = mS. √x∈mS n≥0 xS=mS Furthermore, the following are equivalent:

1. S is archimedean.

2. p = (0).

3. τ < ∞.

4. T is not local.

Proof. In view of Proposition 5.2.5, it is clear that the three deﬁnitions for p are equivalent.

It is clear that (1) ⇒ (2). To see that (2) ⇒ (1), notice that if y ∈ mS, then y ∈ xS √ T n for some x ∈ mS such that xS = mS, so y S ⊂ p; thus if S is non-archimedean, n≥0 p 6= (0). 41

(2) ⇔ (3) follows from Corollary 5.4.3. The implication (3) ⇒ (2) is immediate, and (2) ⇒ (3) follows from the existence of a nonzero y ∈ mS with e(y) > 0, which follows from the assumption dim S ≥ 2. Assuming the negations of (1), (2), and (3), we prove the negation of (4) by √ showing that T = Sp. For a ∈ mS, Proposition 5.0.2.2 implies that aS = mS if and √ only if e(a) = 0. Since τ = ∞, it follows from Corollary 5.4.3 that if aS 6= mS, then a ∈ p. Thus p is the unique maximal ideal in the punctured spectrum of S,

i.e. the unique dimension 1 ideal of S, and T = Sp. The property that p = pT 1 set-theoretically follows from Proposition 5.2.5 and the construction T = S[ x ] for an a mS-primary element x ∈ mS; indeed, if a ∈ p, then xn ∈ p for all n ≥ 0. It remains to be seen that (1), (2), and (3) imply (4). To see this, we make use of

Theorem 5.5.2. Since dim S ≥ 2, there exists an element a ∈ mS with e(a) > 0. Since τ < ∞, also w(a) < ∞, so there exists an element y ∈ T × such that w(y) > w(a). Now consider a and a + y, both of which are elements of T . Then Theorem 5.5.2 implies that e(a + y) = e(a) > 0. Therefore a + y and a are non-units of T , but (a + y) − a = y is a unit of T , proving that T is not local.

5.6 The Complete Integral Closure

Let A be a domain with ﬁeld of fractions F . An element y ∈ F is almost integral over A if there exists an element a ∈ A such that ayn ∈ A for every n ≥ 0. Equiva- lently, y is almost integral over A if the ring A[y] is a fractional ideal of A. The ring A is completely integrally closed if it contains all of the elements in F that are almost integral over it, and the complete integral closure of A is the ring containing all of the elements in F that are almost integral over it. For Noetherian domains, the notion of an element being almost integral is the same as the notion of being integral. For non-Noetherian domains, an element being almost integral is a weaker condition. While a quadratic Shannon extension S is always integrally closed, it often fails to be completely integrally closed. 42

The following proposition classiﬁes the elements that are almost integral over S.

Proposition 5.6.1. Assume notation as in Setting 5.0.1, and assume that S is archimedean of dim S ≥ 2. Let a ∈ K \ S. Then a is almost integral over S if and only if a ∈ T and w(a) = 0.

Proof. Assume that a is almost integral over S, so there exists an element y ∈ S such

n that ya ∈ S for each n ≥ 0. Since a∈ / S, it follows that y ∈ mS, so w(y) > 0 by Proposition 5.2.3.1. As yan ∈ S, we similarly have that w(yan) ≥ 0 for all n ≥ 0,

1 hence w(a) ≥ − n w(y) for all n ≥ 0, so w(a) ≥ 0. Since S ⊆ T , the element a is almost integral over T , so since T is Noetherian and integrally closed, it follows that a ∈ T . Finally, if w(a) > 0, then a ∈ V , hence a ∈ S = T ∩ V , contradicting the assumption that a∈ / S.

Conversely, assume that a ∈ T and w(a) = 0. Let y ∈ S be any mS-primary element. Then w(yan) > 0 for all n ≥ 0, so yan ∈ V . It follows that yan ∈ S = V ∩ T for all n ≥ 0.

This classiﬁcation allows us to describe the complete integral closure S∗ of S.

Theorem 5.6.2. Assume notation as in Setting 5.0.1, and assume that S is archimedean of dim S ≥ 2. Let S∗ denote the complete integral closure of S. Then

∗ S = mS :K mS = T ∩ W.

Proof. Proposition 5.6.1 gives immediately that S∗ = T ∩ W . To see the remaining equalities, recall that by Proposition 5.2.3.1, set-theoretically,

mS = {a ∈ T | w(a) > 0}.

Clearly T ∩ W ⊆ mS :K mS. On the other hand, if a ∈ mS :K mS, then for any n y ∈ mS, it follows by induction on n that ya ∈ mS for all n ≥ 0, proving that ∗ mS :K mS ⊆ S . This completes the proof.

The rational dependence of τ over w(T ×) determines whether or not S is completely integrally closed. Moreover, S is completely integrally closed if and only if V is rank 1. 43

Theorem 5.6.3. Assume notation as in Setting 5.0.1, and assume that S is archimedean of dim S ≥ 2. The following are equivalent:

1. S is completely integrally closed.

2. τ is rationally independent over w(T ×).

3. V = W .

Proof. To see that Item 1 implies Item 2, we prove the contrapositive. Suppose that τ is rationally dependent over w(T ×), say dτ = w(y) for some y ∈ T × and some integer d > 0. Let a ∈ T be any non-unit and let e = e(a) > 0. By Corollary 5.4.3, there exists z ∈ T × such that w(a) = w(z) + eτ. Then

ad w = (dw(a)) − (dw(z) + ew(y)) zdye = (dw(z) + edτ) − (dw(z) + edτ)

= 0

ad ad ∗ and e( zdye ) = de(a) > 0. Thus Proposition 5.6.1 implies that zdye ∈ S \ S, proving the claim. Assume Item 2 and let a ∈ W . If w(a) > 0, then a ∈ V , so assume that w(a) = 0. By Corollary 5.4.3, there exists y ∈ T × such that w(y)+e(a)τ = w(a) = 0,

1 × so e(a)τ = w( y ). Since τ is rationally independent over w(T ), we conclude that e(a) = 0, so a ∈ V by Theorem 5.5.2. That Item 3 implies Item 1 follows from Theorems 5.1.1 and 5.6.2.

5.7 The Non-Archimedean Case

We restate Theorem 5.5.4 for the non-archimedean case by negating all of the equivalent conditions.

Theorem 5.7.1. Assume notation as in Setting 5.0.1. Assume that dim S ≥ 2 and let p = {a ∈ K | w(a) = ∞} be as in Theorem 5.5.4. The following are equivalent: 44

1. S is non-archimedean.

2. p 6= (0).

3. τ = ∞.

4. T is local.

Under these equivalent conditions, p is the unique dimension 1 prime ideal of S,

T = Sp, and the equality p = pT holds set-theoretically.

Proof. Since p consists of all elements of mS that are not mS-primary, it is the unique dimension 1 prime ideal of S. Thus T = Sp. The set-theoretic equation p = pT holds by the construction of T given by inverting all mS-primary elements and by Proposition 5.2.5.

As in the archimedean case, we can explicitly describe the valuation ring V in terms of the functions w and e. In the archimedean case, Theorem 5.5.2 shows that V is determined by first considering the w-value and second the e-value. In the non-archimedean case, the following theorem shows that the reverse happens: V is determined by first considering the e-value and second the w-value. However, in the archimedean case, the function e(−) gives a well-defined integer for all nonzero elements a ∈ K, whereas in the non-archimedean case, the function w(−) yields only ±∞ for elements of K with nonzero e-values. Thus while there exists a canon-

ical (up to ﬁxing an mS-primary element x) valuation ν = (w(−), −e(−)) in the archimedean case, the functions w and e determine no such canonical valuation in the non-archimedean case.

Theorem 5.7.2. Assume notation as in Setting 5.0.1 and assume that S is non- archimedean.

1. e(−) is a rank 1 discrete valuation that deﬁnes a DVR (E, mE).

2. w(−) induces a rank 1 rational rank 1 valuation w(−) that deﬁnes a valuation

ring W on the residue ﬁeld E/mE. 45

3. V is the composite valuation of E and W .

Furthemore, recall that T = (Rn)Pn for n 0 as in Remark 4.4.3. If Pn is a regular

prime ideal for n 0 (i.e. Rn/Pn is a RLR for n 0), then e(−) = ordT .

Proof. First, we show that the multiplicative function e deﬁnes a rank 1 valuation overring of V . By Corollary 5.4.3, for nonzero a ∈ K, if e(a) > 0, then w(a) = ∞,

so a ∈ mV . Similarly, for nonzero a ∈ K and e(a) < 0, then w(a) = −∞, so a∈ / V . Therefore if a ∈ V is nonzero, then e(a) ≥ 0. To see that e deﬁnes a valuation, it suﬃces to show that the set of nonzero elements of K with positive e-value is closed under addition. Let a, b ∈ F be two such elements,

b and assume without loss of generality that (a, b)V = aV . Thus a ∈ V , so e(b) ≥ e(a), b b b b and 1 + a ∈ V , so e(1 + a ) ≥ 0. Thus e(a + b) = e(a(1 + a )) = e(a) + e(1 + a ) ≥ e(a). It follows that e deﬁnes a rank 1 discrete valuation ring E, and the maximal ideal mE of E is a prime ideal of V . This completes the proof of Item 1. Moreover,

mE = {a ∈ K | e(a) > 0} = {a ∈ K | w(a) = ∞} by Corollary 5.4.3. Observe that the valuation ring W induced by w is precisely the boundary valuation ring V . To see this, since W is an overring of V , it suﬃces to show that mV ⊆ mW . Let a ∈ mV and assume by way of contradiction that w(a) = 0. By the ﬁrst item, e(a) = 0. Then Remark 5.3.2 implies that we may write a = ua0 for

× 0 some u ∈ T and some a ∈ V \ mV , so u ∈ mV and w(u) = 0. This contradicts Proposition 5.2.3.4. We conclude that W = V . As w is a possibly inﬁnite valuation, it induces a rank 1 real valuation w on the

quotient ﬁeld of V/{a ∈ mV | w(a) = ∞} and {a ∈ mV | w(a) = ∞} is the unique

dimension 1 prime ideal of V . Since Corollary 5.4.3 implies that mE = {a ∈ mV | w(a) = ∞}, we conclude that the boundary valuation is a rank 2 valuation that is the composite valuation of E and w; cf. [24, 11.3]. Moreover, since S is non-archimedean and τ = ∞ as in Theorem 5.7.1, Proposition 5.4.4 implies that w has rational rank 1. 46

Finally, denote Pn = P ∩Rn for n ≥ 0, and assume in addition that Rn/(P ∩Rn) is a regular local ring for n 0. Then [10, Theorem 10] implies that for every element f ∈ R such that ord (f) = e(f), it follows that ord (f) = ord (f). Since n Rn (Rn)Pn Rn

(Rn)Pn = T , we conclude E is the order valuation ring of T .

5.8 Completely Integrally Closed Examples

By taking a local quadratic transform R0 ( R1, we impose conditions on the

valuation domains that birationally dominate R1. Given some conditions we want to impose on a valuation ring, we can try to construct a sequence of local quadratic

transforms R0 ( ··· ( Rn such that every valuation domain that birationally domi-

nates Rn satisﬁes the desired conditions. We start with a 2-dimensional regular local ring and ﬁxed positive integers N and

k. For every rank 1 valuation ring birationally dominating Rn, after ﬁxing w(m0) = 1, we want to impose the conditions:

n−1 P 1. w(mi) = N + 1, i=0

1 2. w(mi) ∈ Z[ 2 ] for 0 ≤ i < n, and

1 2 3. w(xn) = 2k for some xn ∈ mn \ mn.

Example 5.8.1 shows how to construct a sequence to attain these conditions.

1 Example 5.8.1. Let N be a positive integer and let = 2k for some integer k ≥ 1.

Let (R0, m0) be a 2-dimensional regular local ring with m0 = (x0, y0). We show how

to construct a sequence of blow-ups R0 ( R1 ( ··· ( Rm for some integer m > 0

such that, if w is a rank 1 valuation birationally dominating Rm and w(x0) = 1, then m−1 P w(mi) = N + 1 and w(xm) = . i=0 We start by blowing up in the monomial x-direction n − 1 times. That is, we

yi−1 y0 set xi = x0 and yi = = i for 1 ≤ i ≤ N − 1. Then, for RN , we deﬁne x0 x0 y0 xN = x0 and yN = N − α for some nonzero unit α ∈ R0. Thus, for any rank 1 x0 47

valuation w birationally dominating RN with w(m0) = 1, it follows that w(mi) = 1

y0 for 0 ≤ i ≤ N − 1, and moreover, since w( N ) = 0, we have w(y0) = N. x0 k Next, we blow up in the y-direction m = 2 times. We deﬁne yi = yi−1 and

xi−1 x0 xi = = for N ≤ i ≤ N + m − 1. For RN+m, we define xN+m = xN+m−1 yN yN m yN+m−1 yN and yN+m = − β = − β for some nonzero unit β ∈ R0. Let w be a xN+m−1 x0 m yN rank 1 valuation birationally dominating RN+m with w(m0) = 1. Since w( ) = 0 x0 1 1 and w(x0) = w(m0) = 1, it follows that w(yN ) = m = 2k . Then we have, for 1 N ≤ i ≤ N + m − 1, that w(mi) = 2k . We conclude that N+m−1 N−1 N+2k−1 X X X 1 w(m ) = 1 + = N + 1. i 2k i=0 i=0 i=N 1 and that w(xN+m) = 2k . Next, fix any real number τ > 2. For every rank 1 valuation w birationally dominating a Shannon extension S (and in particular for the asymptotic valuation w as in Definition 5.2.2), after fixing w(m0) = 1, we repeat the construction of Example 5.8.1 to impose the conditions: ∞ P 1. w(mn) = τ, and n=0 1 2. w(mn) ∈ Z[ 2 ] for each n ≥ 0.

Example 5.8.2. Let τ > 2 be an integer. Set τ0 = τ and m0 = 0. To continue the sequence starting from Rmi , use the construction of Example 5.8.1 with N the largest positive integer strictly less than τi − 1 and with any positive integer ki such

ki that 2 (τi − (n + 1)) > 2. Then we obtain a sequence from Rmi to Rmi+1 , and we set

ki τi+1 = 2 (τi − (n + 1)).

In this way, we obtain an inﬁnite sequence {(Rn, mn)}n≥0. Let w be a rank S 1 valuation birationally dominating Rn such that w(m0) = 1. We claim that n≥0 ∞ P w(mi) = τ. To prove this, we show that for any n ≥ 0, i=0 m −1 n−1 n−1 Xn Y 1 Y 1 τ − w(mi) = τn and w(mmn ) = . 2ki 2ki i=0 i=0 i=0 48

We prove the claim by induction on n, where the base case n = 0 is trivial. Suppose the claim is true for n, and let N denote the smallest integer less than or

equal to τn. Then by construction,

mn+1−1 n−1 X Y 1 1 w(mn) = N and τn+1 = τn − N. 2ki 2kn i=mn i=0 Thus

mn+1−1 n−1 X Y 1 τ − w(mi) = (τn − N) 2ki i=0 i=0 n Y 1 = τn+1 . 2ki i=0

n Moreover, by construction, w(x ) = 1 w(m ) = Q 1 , and since w(x ) = mn+1 2kn mn 2ki mn i=0 w(mmn ), we have proved the inductive step.

By choosing the integers kn in such a way that puts an upper bound on τn for

n 0 (for instance, by always choosing kn to be as small as possible), we conclude ∞ P that w(mi) = τ. i=0 Finally, we take the construction in Example 5.8.2 and we adjoin another variable to it. Example 5.8.2 gives a sequence of 2-dimensional RLRs whose union is a rank 1 valuation ring. By adjoining a variable to each of the RLRs in the sequence, we construct a sequence of 3-dimensional RLRs whose union is a 2-dimensional archimedean Shannon extension.

Example 5.8.3. Let τ > 2 be a real number and let {(Rn, mn)}n≥0 be a sequence of local quadratic transforms as in Example 5.8.2. We modify this example by adjoining

an additional variable zn to Rn. Let z0 be an indeterminant over R0, and for n ≥ 0, let z = zn , where w ∈ m is such that m = w . Then define R˜ = R [z ] n+1 wn n n n n n n n (mn,zn) ˜ ˜ and denote mñ the maximal ideal of Rn. Then {(Rn, mñ)}n≥0 is a sequence of local quadratic transforms of 3-dimensional regular local rings. As in the construction in the previous exercise, let w denote the asymptotic order valuation as in Defini- 49

tion 5.2.2 and ﬁx w(m˜0) = 1. Note that w restricts to a rank 1 valuation on the ﬁeld of fractions of the original Rn. Thus we have

∞ ∞ X X w(m˜n) = w(mn) = τ. n=0 n=0 Notice that e(z ) = ord (z ) = 1, so w(z ) = τ. 0 R˜0 0 0

S The resulting Shannon extension S = Rn of Example 5.8.3 is 2-dimensional n≥0 and archimedean, and it is completely integrally closed if and only if τ is irrational.

× 1 This follows from Theorem 5.6.2 and from the fact w(T ) = Z[ 2 ]. If τ is irrational, then τ is rationally independent over w(T ×), S is completely integrally closed, w has rational rank 2, and V = W . If τ is rational, then τ is rationally dependent over

w(T ×), S is not completely integrally closed, w has rational rank 1, and V ) W . 50

6. LIMITS OF REES VALUATIONS

In Chapter 5, we proved a number of strong results for the union of an inﬁnite

sequence {(Rn, mn}n≥0 of local quadratic transforms of regular local rings. Among

those results, we proved that the sequence of order valuations {ordRn }n≥0 converges in the patch topology. In the present chapter, we drop the regularity condition and show that this theorem holds in a more general setting where the rings Rn are Noetherian local domains.

In order to prove a generalization, we must replace the order functions ordRn ,

which are not valuations in general, with the ﬁnite set Rees mn of Rees valuations

of mn. This is a direct generalization: if Rn is a regular local ring, then Rees mn =

{ordRn }. In order for there to exist an analogous sequence of Rees valuations, we

must make a strong assumption: we assume that mn has a unique Rees valuation, say

Rees mn = {νn}, for sufficiently large n. After making another minor assumption and modification that each has no effect in the regular setting, we prove that the sequence

of Rees valuations {νn}n≥0 converges in the patch topology.

6.1 Normalizing Local Quadratic Transforms

In Chapter 5, we made extensive use of local quadratic transforms: every local quadratic transform of a regular local ring is itself a regular local ring. However, a local quadratic transform of an integrally closed Noetherian local rings may fail to be integrally closed in general. To apply the properties of normality, we make the following deﬁnition:

Deﬁnition 6.1.1. Let (R, m) be a Noetherian local domain. A local normalized

quadratic transform of R is a local ring (T, mT ) on the normalized blow-up ProjR[mt] of m. 51

A local normalized quadratic transform (T, mT ) of R dominates a unique local quadratic transform (S, mS) of R. The ring T can be obtained from S by taking the integral closure S of S and localizing at a maximal ideal lying over mS. There is a problem: it is well known that the integral closure of a Noetherian domain need not be module-ﬁnite or even Noetherian. In [24, Appendix A1], ”Examples of bad Noetherian rings,” Nagata gives an example of a 3-dimensional Noetherian ring whose integral closure is not Noetherian. To guarantee that every local normalized quadratic transform is a Noetherian local domain, we impose an additional condition:

Deﬁnition 6.1.2. A Noetherian local ring (R, m) is analytically unramiﬁed if its completion Rˆ is reduced and analytically irreducible if Rˆ is a domain.

Theorem 6.1.3. [26, Theorems 1.2 and 1.5] A Noetherian local domain (R, m) is analytically unramified if and only if for every module-finite birational extension S of R, its integral closure S is a finite S-module.

We lose little generality in assuming that a Noetherian local domain (R, m) is analytically unramified. It a weak condition, satisfied by the Noetherian local domains that arise in algebraic geometry; for instance, it is satisfied if R contains a field.

Furthermore, we need only assume that R0 is analytically unramiﬁed, because Rees’s

theorem implies that every Noetherian local ring (Rn, mn) obtained by iterated local

normalized quadratic transforms of (R0, m0) is itself analytically unramiﬁed.

6.2 Rees Valuations

Let R be a Noetherian domain with ﬁeld of fractions K and let I ⊂ R be a nonzero ideal. Then associated to I, there is a nonempty ﬁnite set Rees(I) of discrete rank 1 valuations birational over R called the Rees valuations of I. The Rees valuations play an important role in both the asymptotic properties of powers of the ideal I and in the properties of the blow-up ring of I. We give two equivalent characterizations of the Rees valuations of I: 52

1. Rees(I) is the unique minimal set of valuations birational over R that determine the integral closure for every power of I.

2. Rees(I) correspond to minimal primes of IProjR[It], the normalized blow-up of I.

Definition 6.2.1. [20, Definition 10.1.1] Let R be a Noetherian domain with field of fractions K and let I ⊂ R be a nonzero ideal. A set of discrete valuations {ν1, . . . , νr} of K is the set Rees I of Rees valuations of I if

n Tr n 1. I = i=1 I Vr for all n ≥ 1, where Vi denotes the valuation ring of νi.

2. The set {ν1, . . . , νr} is minimal with respect to the ﬁrst condition.

Every nonzero ideal I ⊂ R has a nonempty unique ﬁnite set of Rees valuations [20, Theorem 10.1.6], and they arise naturally on the blow-up of I.

Theorem 6.2.2. [20, Theorem 10.2.2] Let R be a Noetherian domain, let I =

I (a1, . . . , ar)R be an ideal, let Si = R[ ] be an aﬃne chart of the blow-up of I, and ai

let Si be the integral closure of Si. The Rees valuation rings of I are precisely the

localizations (Si)p at prime ideals p minimal over ISi.

The order function of the ideal I, after taking an asymptotic limit, is completely determined by the Rees valuations of I. Deﬁne the asymptotic order function of I by

n ˜ ordI (a ) ordI (a) = lim . n→∞ n ˜ If the associated graded ring grI (R) is reduced, then ordI (a) = ordI (a). The asymptotic order function of I is equal to the minimum value over the Rees valuation, normalized for the ideal I:

Theorem 6.2.3. [20, p. 192] Let Rees I = {ν1, . . . , νn}. Then for a ∈ R, ˜ νi(a) ordI (a) = min | i = 1, . . . , n . νi(I)

If (R, m) is a regular local ring, then Rees m = {ordR} [20, Example 10.3.1]. 53

6.3 Multiplicity and Blowing Up

Let (R, m) be a d-dimensional Noetherian local domain, let I ⊂ R be an m-primary ideal, and let M be a ﬁnite d-dimensional R-module. Then the multiplicity of M with respect to I, denoted e(I,M), is λ (M/InM) e(I,M) = lim R , n→∞ dim M

where λR(−) denotes length as an R-module. We define the multiplicity of I to be e(I) = e(I,R) and we define the multiplicity of R itself to be e(R) = e(m). The multiplicity of M with respect to I arises as the normalized leading coefficient of the Hilbert-Samuel polynomial of M with respect to I, and is thus a positive integer. Multiplicity is a classical construction that’s been extensively studied. Nagata proves [24, Theorem 40.6] that for an unmixed Noetherian local ring, regularity is equivalent to multiplicity 1.

Theorem 6.3.1. [24, Theorem 40.6] A Noetherian local ring (R, m) is regular if and only if R is unmixed1 and e(R) = 1.

In this sense, the multiplicity of a Noetherian local ring is a coarse measure of how far it is from being regular, and one can naively try to resolve a singularity by searching for a method to decrease multiplicity. Indeed, one can attempt to ”improve” multiplicity by blowing up. Bennett [3, Theorem (0)] and Hironaka [12, Theorem I] deﬁne a class of permissible blow-ups and show that for this class, while multiplicity need not strictly decrease under blow-up, at least multiplicity cannot ”worsen.” The class of permissible blow-ups includes local quadratic transforms:

Proposition 6.3.2. [12, Theorem I] Let (R0, m0) be a local quadratic transform of (R, m). Then e(R) ≥ e(R0).

Multiplicity is especially well-behaved for ﬁnite birational extensions. We state a well known extension formula for multiplicity:

1A Noetherian local ring (R, m) is unmixed if dim Rˆ = dim R/ˆ p for every associated prime p ∈ Ass(Rˆ) in the completion Rˆ of R. 54

Proposition 6.3.3. [20, Theorem 11.2.7] Let S be a module-ﬁnite birational extension of R and let q ⊂ R be an m-primary ideal. Then

X eR(q) = eSn (qSn)[S/n : R/m]. n∈MaxS dim Sn=dim R Under the notation of the previous proposition, let n denote a maximal ideal of

S with dim Sn = dim R. Since mSn ⊂ nSn, it follows that eSn (mSn) ≥ e(Sn). Since

e(R) ≥ eSn (mSn) by the proposition, it follows that e(R) ≥ e(Sn). We record this statement in a corollary.

Corollary 6.3.4. If S is a localization at a maximal ideal of a module-ﬁnite birational extension of R with dim S = dim R, then e(S) ≤ e(R).

e(R) = eR(m) ≥ eT (mT )[T/mT : R/m] ≥ e(T ).

By taking a normalized local quadratic transform, we are taking a local quadratic transform, then taking the localization of a module-ﬁnite birational extension. Thus it follows from Proposition 6.3.2 and Cororollary 6.3.4:

Proposition 6.3.5. Let (R, m) be an analytically unramiﬁed Noetherian local domain

and let (R1, m1) be a local normalized quadratic transform of R with dim R = dim R1.

Then e(R1) ≤ e(R).

m Let p be a height 1 prime ideal of R, let A = R[ x ] be an aﬃne chart of the blow-up ProjR[mt], and consider a height 1 prime ideal q of A lying over a height 1 prime ideal p of R. Notice that x∈ / p, since xA = mA lies over m. Since A is birational over R and Rp is a DVR, it follows that Aq = Rp, so by permutability of localization and residue class formation, A/q is birational over R/p. Let m = (x, y1, . . . , yr) and

yi let R[T1,...,Tr] A be a presentation, where Ti maps to x . Then consider the composition with the quotient map A A/q, call this composition φ. Certainly

pR[T1,...,Tr] ⊂ ker φ, so φ induces a map (R/p)[T1,...,Tr] A/q. Moreover, the image of Ti in the domain A/q satisﬁes the relation φ(Ti)x = yi. Since x 6= 0, it 55

yi follows that φ(Ti) = x in the quotient field of A/q, which equals the quotient field of R/p. We conclude that φ is a presentation of the affine chart of the blow-up of the maximal ideal m of R/p, and in particular, A/q is an affine chart of the blow-up of m of R/p. Let (R0, m0) be a local quadratic transform of R with dim R = dim R0 and let p0 be a height 1 prime ideal of R0 that lies over a height 1 prime p of R. By the preceding paragraph, it follows that R0/p0 is a local quadratic transform of R/p. Assume in addition that R is analytically unramified and universally catenary (for instance, if m

is one-fibered) and let (R1, m1) be a local normalized quadratic transform of R such 0 that R1 dominates R . Let p1 be the unique height 1 prime ideal of R1 lying over p. 0 Since R1 is a localization of a finite R -module, it follows that R1/p1 is a localization 0 0 0 of a finite (R /p )-module. Since R1 to R is a birational extension and (R1)p1 is a 0 DVR, it follows that (R1)p1 = Rp0 , so once again, by permutability of localization and 0 0 residue class formation, the extension R1/p1 to R /p is birational. The dimension formula for universally catenary Noetherian domains implies that dim R0 = dim R,

0 0 0 and since R1 and R are universally catenary, dim R /p = dim R1/p1 = dim R − 1.

Then Corollary 6.3.4 implies that e(R1/p1) ≤ e(R/p). We record this as a lemma to our main theorem.

Lemma 6.3.6. Let R be a universally catenary analytically unramiﬁed Noetherian

local domain. Let (R1, m1) be a normalized local quadratic transform of (R, m) with

dim R1 = dim R and let p1 be a height 1 prime of R1 lying over a height 1 prime p of R. Then

e(R1/p1) ≤ e(R/p).

6.4 Degree Functions

Associated to an m-primary ideal q of a Noetherian local domain (R, m) is the

degree function of q, an additive function dq : R \{0} → Z≥0 that assigns a positive

value to every nonzero nonunit of R. The degree function dq is a weighted sum of 56 the Rees valuations of q, and if the ring R is normal, it is determined on an element x ∈ R by the prime divisors of R. If R is a regular local ring and q = m, then the degree function simpliﬁes to dm = ordR.

Deﬁnition 6.4.1. Let (R, m) be a Noetherian local ring and let q be an m-primary ideal. The degree function of q, denoted dq(−), is the function

dq : R \{0} → Z≥0

dq(x) = eR/xR(q + xR/xR)

If in addition R is normal, we deﬁne the degree function of q on a height 1 prime p of R to be

dq(p) = eR/p(q + p/p).

Rees shows that dq is equal to a weighted sum of the Rees valuations of q.

Theorem 6.4.2 ( [25, Theorem 2.3]). Let (R, m) be a Noetherian local domain, let q be an m-primary ideal, and let Rees q = {ν1, . . . , νr}. Then there are positive integer constants c1, . . . , cr such that for all nonzero x ∈ R, r X dq(x) = ciνi(x). i=1 Recall that if R is a normal Noetherian local domain, then for nonzero x ∈ R, the principal ideal xR has a primary decomposition of the form

(νp ) 1(x) (νps (x)) xR = p1 ··· ps , where νp denotes the p-adic valuation. In the same paper, Rees notes that if in addition R is normal, then the associative formula for multiplicity applied to the primary decomposition of xR implies that the degree function is completely determined by the values of the height 1 prime ideals of R.

Theorem 6.4.3 ( [25, p. 5]). In addition to the hypotheses of Theorem 6.4.2, assume that R is normal. Let Σ denote the set of height 1 prime ideals of R and let νp denote the discrete valuation associated to a height 1 prime p. Then for all nonzero x ∈ R, X dq(x) = νp(x)dq(p). p∈Σ 57

Putting these together, we obtain the equation relating the Rees valuations of q and the height 1 primes of R, r X X dq(x) = ciνi(x) = νp(x)dq(p). (6.1) i=1 p∈Σ

6.5 Convergence of Rees Valuations

We ﬁx the following notation:

Setting 6.5.1. Let (R0, m0) be an analytically unramiﬁed normal Noetherian local domain, and for all n ≥ 0, let (Rn+1, mn+1) be a local normalized quadratic transform of (Rn, mn). By replacing R0 with Rn for some suﬃciently large n, assume that d := dim R0 = dim Rn for all n ≥ 0.

Theorem 6.5.2. Assume the notation of Setting 6.5.1. If mn is one-ﬁbered for n 0, then the corresponding sequence of Rees valuations converges.

Proof. Replace R0 with Rn for a suﬃciently large n to assume that mn is one-ﬁbered for all n ≥ 0. Let νn denote the unique Rees valuation of mn and denote by dn(−) the degree function of mn on Rn. By Theorem 6.4.2, there is an integer constant cn > 0 such that dn(x) = cnνn(x) for all x ∈ Rn.

Let f, g ∈ R0. We aim to show that the sign of νn(f)−νn(g) is constant for n 0.

Write primary decompositions for fRn and gRn, say r s \ (αi) \ (βi) fRn = pi , gRn = qi . i=1 i=1

Re-arrange the pi and qj such that pi = qi for i ≤ t and the pi, qi are all distinct for i > t. Then, as in Proposition 6.4.3, r s X X cnνn(f) = dn(f) = αidn(pi), cnνn(g) = dn(g) = βidn(qi). i=1 i=1 Then there is a ”common part” of f, g, namely X Dn = min{αi, βi}dn(pi). 1≤i≤t 58

Denote an and bn to be the reduced sums,

t r X X an = cnνn(f) − Dn = (αi − min{αi, βi})dn(pi) + αidn(pi), i=1 i=t+1

t s X X bn = cnνn(g) − Dnbn = (βi − min{αi, βi})dn(pi) + βidn(qi). i=1 i=t+1

Suppose without loss of generality that νn(f) ≤ νn(g), or equivalently, that an ≤

bn. We claim that an+1 ≤ an.

To see this claim, consider the primary decompositions of fRn+1 and gRn+1. Let 0 0 d denote the center of the Rees valuation νn on Rn+1. Denote pi (qi) to be the unique lift of pi (qi) to Rn+1 if it exists and the ring Rn+1 itself otherwise. Then

r s νn(f) \ 0 (αi) νn(g) \ 0 (βi) fRn+1 = d ∩ (pi) , gRn+1 = d ∩ (qi) . i=1 i=1 And again, r X 0 cn+1νn+1(f) = νn(f)dn+1(d) + αidn+1(pi), i=1 s X 0 cn+1νn+1(g) = νn(g)dn+1(d) + βidn+1(qi). i=1

where dn+1(Rn+1) = 0. 0 0 Lemma 6.3.6 implies that dn+1(pi) ≤ dn(pi) and dn+1(qi) ≤ dn(qi) for all i. By

assumption, νn(f) ≤ νn(g), so νn(f) − min{νn(f), νn(g)} = 0. Thus

t r X 0 X 0 an+1 = (αi − min{αi, βi})dn+1(pi) + αidn+1(pi) i=1 i=t+1 t r X X ≤ (αi − min{αi, βi})dn(pi) + αidn(pi) i=1 i=t+1

= an.

Thus we have shown that {min{an, bn}}n≥0 is a decreasing sequence of nonnegative integers, so it stabilizes to some nonnegative integer value for n 0, say for n ≥ N.

Suppose without loss of generality that aN ≤ bN . Then an = aN and an ≤ bn for

all n ≥ N. If am = bm for any m ≥ N, then bn = bm for all n ≥ m, so an = bn for 59

all n 0. In any case, the sign of an − bn is eventually constant, hence the sign of vn(f) − vn(g) is eventually constant. 60

7. MONOMIAL LOCAL QUADRATIC TRANSFORMS

Let (R, m, k) be a d-dimensional regular local ring and let X = ProjR[mt] denote the blow-up of m. Let x ∈ m \ m2, so xR generates a height 1 regular prime, and consider

m d−1 the affine chart A = R[ x ] of X. The fiber of m in A is isomorphic to Ak ; that is, if ∼ y z m = (x, y, . . . , z) is a regular system of parameters, then A/m = k[ x ,..., x ]. There are many d-dimensional local quadratic transforms of R that are localizations of A. By fixing a regular system of parameters m = (x, y, . . . , z), we can uniquely associate the d-dimensional monomial local quadratic transform of R in the x-direction,

x m m R := R[ ]x, y ,..., z = R[ ]x ,y ,...,z . x x x x 1 1 1

w Here we denote x1 = x, and for each regular parameter w 6= x, we denote w1 = x , so x R = R[y1, . . . , z1]x1,y1,...,z1 . Every local quadratic transform of R with no residue extension is monomial in

some sense. If (R1, m1, k1) is a local quadratic transform of R, then R1 is a monomial local quadratic transform of R with respect to some regular system of parameters

if and only if k = k1. If k is algebraically closed, then every d-dimensional local quadratic transform is monomial. The utility of monomial local quadratic transforms is not for considering a sin-

gle local quadratic transform R ⊂ R1, but rather for examining a sequence of successive monomial local quadratic transforms. Fix a regular system of parameters

m = (x, y, . . . , z). If (R1, m1) is a monomial local quadratic transform of R, then

there is an induced regular system of parameters m1 = (x1, y1, . . . , z1). We say a

(ﬁnite or inﬁnite) sequence {(Rn, mn)}n≥0 of local quadratic transforms is monomial

with respect to a regular system of parameters on R = R0 if for all n, Rn+1 is a

monomial local quadratic transform of Rn with respect to the induced regular system

of parameters on Rn. 61

In this chapter, we explore sequences of monomial local quadratic transforms.

7.1 Monomial Valuations

The simplest valuations to describe are the monomial valuations. Given a regular local ring (R, m), one deﬁnes a monomial valuation with respect to a ﬁxed regular system of parameters m = (x, y, . . . , z)R by assigning nonnegative real values ν(x) =

αx, ν(y) = αy, . . . , ν(z) = αz. Then for every other element f ∈ R, we deﬁne f to have the minimal possible value deﬁned by these constraints. Since these values

must extend additively to monomials, for a monomial m = xγx yγy ··· zγz , its value is

ν(m) = γxαx + γyαy + ··· + γzαz. Then for an arbitrary element f ∈ R, we may write P f = i uimi as the sum of distinct monomials mi with unit coefficients ui, and we define ν(f) = mini ν(mi). Usually, one considers monomial valuations for a polynomial ring over a field, where every element can be written as such a sum of monomials in a unique way. In this chapter, we shall consider general monomial valuations defined for a regular system of parameters of a regular local ring. While the representation of an element f ∈ R as a sum of monomials is far from unique, the value ν(f) is uniquely defined. See [21] for more information on monomial valuations defined on a regular local ring and see [21, Proposition 3.2] for a proof that ν(f) is independent of representation. Let (R, m) be a d-dimensional regular local ring and let V be a monomial valuation birationally dominating R with respect to a regular system of parameters

m = (x1, . . . , xd). Assume without loss of generality that ν(x1) ≤ ν(x2) ≤ ... ≤ ν(xd). If all variables have the same value, then V is the order valuation ring of R. Other- wise, the ﬁrst r variables have the same value for some 1 ≤ r < d. Let (R0, m0) denote the local quadratic transform of R along V . Then

0 x2 xd R = R[ ,..., ] xr+1 xd , (x1, ,..., ) x1 x1 x1 x1 where dim R0 = d − (r − 1). We take this to be our formal deﬁnition for a monomial local quadratic transform: 62

Deﬁnition 7.1.1. The ring R0 deﬁned here is the monomial local quadratic transform

of R in the (x1, . . . , xr)-direction with respect to the regular system of parameters 0 m = (x1, . . . , xd). If r = 1, then R is the monomial local quadratic transform in the

x1 0 x1-direction and we denote R := R .

0 If x1 has the unique minimal V -value, then the dimension of R is the same as the dimension of R. Otherwise, the dimension of R0 is strictly less than the dimension of

R, and R0 is a proper localization of the rings Rx1 ,...,Rxr . Notice that V is a monomial valuation with respect to the induced regular system

0 xr+1 xd of parameters m = (x1, ,..., ). By iterating, we conclude: x1 x1

Remark 7.1.2. Let (R, m) be a regular local ring and let V be a monomial valuation

ring on R. Then the sequence of local quadratic transforms {Rn}n≥0 of R along V is monomial.

Recall that this sequence is inﬁnite if and only if V is not a prime divisor of R. It is simple to detect whether a monomial valuation is divisorial:

Remark 7.1.3. For a valuation ring V dominating R induced by a monomial valuation, V is divisorial ⇔ V is a DVR ⇔ V has rational rank 1.

Proof. If V is a prime divisor, then V is a DVR, and if V is a DVR, then V has rational rank 1. It remains to show that if V has rational rank 1, then V is a prime divisor. To see this, notice that if V has rational rank 1, we may clear denominators to assume ν(x), ν(y), . . . , ν(z) are integers. Consider the sequence

{Rn} of local quadratic transforms along V . If ν(xn) = ν(yn) = ... = ν(zn),

then V is the order valuation ring of Rn and the sequence terminates. Otherwise,

max{ν(xn+1), ν(yn+1), . . . , ν(zn+1)} ≤ max{ν(xn), ν(yn), . . . , ν(zn)} − 1. Since this is a strictly decreasing sequence of positive integers, the sequence must terminate, so V is a prime divisor. 63

In the 2-dimensional case, sequences of local quadratic transforms along a monomial valuation are especially simple. Let m = (x, y) and define a monomial valuation ν by ν(x) = 1 and ν(y) = α for some positive irrational real number α. Then the sequence of monomial local quadratic transforms of R along this valuation corresponds to the continued fraction representation of α. By Abhyankar’s result [1, Lemma 12], S the union Rn is the valuation ring defined by ν. n≥0 The 3-dimensional case is already far more complicated. Let m = (x, y, z) and assign the values ν(x) = 1, ν(y) = α > 1, and ν(z) > 1 + α, where α is irrational. Then the condition ν(z ) > ν(x ) + ν(y ) holds for all n ≥ 0. Since neither zn n n n xnyn nor xnyn are in the union S R , it is not a valuation ring. Moreover, the valuation zn n n≥0 cannot be recovered from the sequence. In fact, we obtain the same induced sequence of local quadratic transforms along any rank 1 valuation with the same properties, even one that is not monomial. A (finite or infinite) sequence of d-dimensional monomial local quadratic transforms corresponds to a sequence of monomials in the obvious way. For a d-dimensional regular local ring (R, m), fix a regular system of parameters m = (x, y, . . . , z). Let

t M = {un}n=0 (with t ∈ N ∪ {∞})) be a sequence of monomials, where un ∈ {x, y, . . . , z}. Then there is an associated sequence of d-dimensional monomial lo-

t+1 cal quadratic transforms {(Rn, mn)}n=0, where Rn+1 is the monomial local quadratic

un transform of Rn in the un-direction, i.e. Rn+1 = Rn . The correspondence goes both ways; given a sequence of d-dimensional monomial local quadratic transforms

t+1 t {(Rn, mn)}n=0, there is an associated sequence of monomials M ∈ {x, y, . . . , z} .

7.2 Conditions for the union to be a valuation

For a general sequence {(Rn, mn)}n≥0 of local quadratic transforms of regular S local rings, we discussed in Chapter 4 conditions such that the union S = Rn n≥0 is a valuation ring. Recall that the ring S is a rank 1 valuation ring if and only if the sequence switches strongly inﬁnitely often; that is, S is a rank 1 valuation 64

ring if and only if for every n ≥ 0 and every f ∈ mn, there exists m > n such

(Rm) that ordRm ((fRn) ) < ordRn (fRn). For a sequence of monomial local quadratic transforms, this condition is easy to detect from the corresponding sequence M of monomials. In the following lemma, we ﬁrst consider the case of ﬁnite sequences.

t Lemma 7.2.1. Let {(Rn, mn)}n=0 be a ﬁnite sequence of d-dimensional monomial local quadratic transforms with respect to m = (x, y, . . . , z), where 1 ≤ t < ∞, and let

t−1 M = {un}n=0 be the associated sequence of variables. The following are equivalent:

1. Each variable w appears at least once among the un.

(Rn) 2. For each f ∈ m0, ordRn ((fR0) ) < ordR0 (f).

Proof. Let mn = (xn, yn, . . . , zn) denote the induced regular system of parameters on

Rn.

(Rn+1) To see (2) ⇒ (1), let w be a variable. Then for 0 ≤ n < t,(wnRn) is either Rn+1, if Rn+1 is in the w-direction, or wn+1Rn+1 otherwise. Thus by induction,

(Rt) Rt (w0R0) = Rt if un = w for some 0 ≤ n < t and (w0R0) = wtRt, otherwise. Since

(Rt) (w0R0) = Rt by assumption, un = w for some 0 ≤ n < t. To see (1) ⇒ (2), we consider the associated graded rings of the regular local rings ∼ Rn. For 0 ≤ n ≤ t, write grmn (Rn) = k[xn, yn,..., zn] for the associated graded ring of Rn, which is a polynomial ring in d variables over the ﬁeld k = R0/m0 = Rt/mt, and write inRn (f) for the initial form of f in grmn (Rn). Denote w = un, so the local quadratic transform from Rn to Rn+1 is in the w-direction. Let f ∈ mn, and denote

0 −r 0 0 (Rn+1) 0 r = ordRn (f), f = wn f, and r = ordRn+1 (f ). By deﬁnition, fRn = f Rn+1. We prove the following about the initial forms of f and f 0:

0 Claim 1: If the variable w appears in inRn (f), then r < r. 0 Claim 2: If r = r, then every variable that appears in inRn (f) also appears in 0 inRn+1 (f ).

These two claims will complete the proof by the following argument. Given f0 ∈

R0, let f1, . . . , ft denote generators for the successive transforms of f0R0 in R1,...,Rt. 65

Since inR0 (f) 6= 0, some variable v appears in in(f0), and by assumption, for some 0 ≤

n < t, the blow-up from Rn to Rn+1 is in the v-direction. If ordRn (fn) < ordR0 (f0), then we are done. Otherwise, by Claim 2 and induction, the variable v also appears in in(fn). Then Claim 1 implies that ordRn+1 (fn+1) < ordRn (fn) and the proof is complete.

e To prove Claim 1, let wn be the highest power of wn that appears in in(f) and e write in(f) = wn m + g, where m is a k-linear sum of monomials of degree r − e in

the remaining variables and where g is a degree r polynomial that has wn-degree at

most e − 1. Lift the elements m and g to Rn by lifting the coeﬃcients of k to units k of Rn to obtain a representation of f, say f = wnm + g + h, where ordRn (h) > r. 0 −r −r −r Now consider f = wn f = m + wn g + wn h. By construction, ordRn+1 (m) = e and −r −r ordRn+1 (wn g) > e. If ordRn+1 (wn h) 6= e, we are done by the subadditive property of

valuations, so assume equality. By construction, inRn+1 (m) is not divisible by wn+1. r+1 0 0 On the other hand, since ordRn (h) > r, we may write h = wn h for some h ∈ Rn+1, 0 so it follows that inRn+1 (h) = inRn+1 (wnh ) is divisible by wn+1. Since the initial −r 0 forms of m and wn h in Rn do not cancel, we conclude that ordRn+1 (f ) ≤ e < r. This completes the proof of Claim 1. To prove Claim 2, assume that r0 = r. Write in(f) = m, where m is a k-linear sum of monomials. Now lift m by lifting the coeﬃcients of k to units of Rn to

obtain a representation f = m + h, where ordRn (h) > r. By Claim 1, the variable r 0 0 w does not appear in m. Thus we may write m = wnm , where m is the sum of

the corresponding variables of Rn+1 with the same coeﬃcients, and so it follows that 0 0 −r 0 f = m + wn h. By the assumption that r = r and the subadditive property of −r −r valuations, ordRn+1 (wn h) ≥ r. If ordRn+1 (wn h) > r, it does not contribute to the 0 −r initial form of f in Rn+1 and we are done, so assume that ordRn+1 (wn h) = r. As in r+1 0 0 Claim 1, we may write h = wn h for some h ∈ Rn+1, so as in Claim 1, the initial −r −r form of wn h in divisible by wn+1. Thus none of the terms of inRn+1 (wn h) cancel 0 with any terms of inRn+1 (m ), so we are done. This completes the proof of Claim 2. 66

The following theorem is immediate from the lemma:

Theorem 7.2.2. Let {Rn}n≥0 be an inﬁnite sequence of d-dimensional monomial local quadratic transforms with respect to m = (x, y, . . . , z) and let M be the corre- S sponding sequence of variables. Then the union Rn is a rank 1 valuation ring if n≥0 and only if each variable appears an inﬁnite number of times in M.

In addition, one can detect whether or not the union of such a sequence is a rank 1

valuation ring by considering the multiplicity sequence. For a sequence {(Rn, mn)}n≥0 of local quadratic transforms along a rank 1 valuation ν, its multiplicity sequence is P {ν(mn)}n≥0, and we previously considered its sum τ = ν(mn). If there are elements n≥0 f1, . . . , ft ∈ m0, t ≥ 2, whose values ν(f1), . . . , ν(ft) are rationally independent, then we proved in Proposition 5.4.4 an explicit upper bound for τ, namely

ν(f ) + ··· + ν(f ) τ ≤ 1 t . t − 1

For a rational rank d monomial valuation ν with respect to m = (x, y, . . . , z), the values ν(x), ν(y), . . . , ν(z) are rationally independent, so the upper bound is

ν(x) + ν(y) + ··· + ν(z) τ ≤ . d − 1

In the following theorem, we prove that we attain this upper bound if and only if the union is equal to the valuation ring of ν.

Theorem 7.2.3. Let {(Rn, mn)}n≥0 be the sequence of local quadratic transforms along a rational rank d monomial valuation ν with respect to m = (x, y, . . . , z) and P S let τ = ν(mn) be the sum of the multiplicity sequence. Then the union Rn is n≥0 n≥0 ν(x) + ν(y) + ··· + ν(z) the valuation ring for ν if and only if τ = . d − 1

P ν(xn)+ν(yn)+···+ν(zn) Proof. For n ≥ 0, denote τn = ν(mn) and σn = d−1 . i≥n 67

Assume without loss of generality that Rn to Rn+1 is in the x-direction, so ν(xn) = ν(m ). Thus x = x and for each remaining variable w, w = wn . Then n n+1 n n+1 xn ν(x ) + (ν(y ) − ν(m )) + ··· + (ν(z ) − ν(m ) σ = n n n n n n+1 d − 1 ν(x ) + ν(y ) + ··· + ν(z ) − (d − 1)ν(m ) = n n n n d − 1

= σn − ν(mn).

n−1 P By induction, σn = σ0 − ν(mi), so τ0 − σ0 = τn − σn. In particular, the equality i=0 τ0 = σ0 holds as in the theorem if and only if τn = σn holds for any n ≥ 0, and we may replace R0 with Rn for any n ≥ 0. S P Assume that Rn is the valuation ring of ν. Since ν(mn), an inﬁnite sum n≥0 n≥0 of positive real numbers, is bounded, it follows that lim τn = 0 and lim ν(mn) = 0. n→∞ n→∞

Since for each variable w, {ν(wn)}n≥0 is a nonincreasing sequence and ν(mn) = ν(wn)

for inﬁnitely many n by Theorem 7.2.2, it follows that lim ν(wn) = 0 for all variables n→∞

w. Therefore lim σn = 0. Since both lim τn = 0 and lim σn = 0, it follows that n→∞ n→∞ n→∞

lim (τn − σn) = 0. But τn − σn is constant, so τ0 − σ0 = 0 and we have proved the n→∞ claim.

Assume that τ0 = σ0, so τn = σn for all n ≥ 0. Replace R0 with Rn so that each variable either appears inﬁnitely many times in M or does not appear at all.

Since lim τn = 0, it follows that lim τn = 0, so lim ν(wn) = 0 for each variable w. n→∞ n→∞ n→∞ Assume by way of contradiction that there exists a variable w that does not appear n−1 P in M. Thus ν(w0) = ν(mn) + ν(wn). Taking the limit as n → ∞, it follows that i=0 ν(w0) = τ0 + 0 = τ0. Taking the equation τ0 = σ0 and substituting ν(w0) for τ0, we obtain the expression

ν(x0) + ν(y0) + ··· + (2 − d)ν(w0) + ··· + ν(z0) = 0.

This contradicts the rational independence of ν(x0), ν(y0), . . . , ν(z0), so we have de- S rived a contradiction. We conclude that no such variable exists, so Rn is the n≥0 valuation ring of ν by Theorem 7.2.2. REFERENCES 68

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