Finitely supported ⁎-simple complete ideals and multiplicities in a regular local ring

Journal of Algebra 488 (2017) 290–314

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Finitely supported ∗-simple complete ideals and multiplicities in a regular local ring Mee-Kyoung Kim Department of Mathematics, Sungkyunkwan University, Suwon 16419, Republic of Korea

a r t i c l e

i n f o

Article history: Received 6 October 2015 Available online 27 June 2017 Communicated by Bernd Ulrich MSC: primary 13A30, 13C05 secondary 13E05, 13H15 Keywords: Rees valuation Finitely supported ideal Special ∗-simple complete ideal Base points Point basis Transform of an ideal Local quadratic transform

a b s t r a c t Let (R, m) and (S, n) be regular local rings of dim(S) = dim(R) ≥ 2 such that S birationally dominates R, and let V be the order valuation ring of S with corresponding valuation ν := ordS . Assume that I S = S and ν ∈ ReesS I S . Let u := αt with IS = αI S , where α ∈ S. Then V = W ∩ Q(R) with W = (R[It])Q = (S[I S u])Q , where Q ∈ Min(mR[It]) and Q ∈ Min(nS[I S u]). Let P, P  be the center of W on S R R[It] and S[I s u], respectively. We prove that if [ n : m ] = 1, s

then R[It] = S[IP  u] . Let I be a finitely supported complete P m-primary ideal of a regular local ring (R, m) of dimension d ≥ 2. Let T be a terminal base point of I and V be the mT -adic order valuation of T with corresponding valuation v := ordT . Let n ≥ 1 be an integer. Assume that I T = mn T and R [ mT : m ] = 1. Let P ∈ Min(mR[It]) such that P = Q ∩ R[It] T

with V = (R[It])Q ∩ Q(R), where Q ∈ Min(mR[It]). We prove that the quotient ring R[It] is d-dimensional normal P Cohen–Macaulay standard graded domain over k with the multiplicity nd−1 . In particular, R[It] is regular if and only if P R n = 1. We prove that k := m is relatively algebraically closed in kv :=

E-mail address: [email protected]. http://dx.doi.org/10.1016/j.jalgebra.2017.06.012 0021-8693/© 2017 Elsevier Inc. All rights reserved.

V mV

. Also we determine the multiplicity of

R[It] , P

and

M.-K. Kim / Journal of Algebra 488 (2017) 290–314

291

we prove that if I T = mT , then R[It] is regular if and only if P R [ mT : m ] = 1. T © 2017 Elsevier Inc. All rights reserved.

1. Introduction All rings we consider are assumed to be commutative with an identity element. We use the concept of complete ideals as defined and discussed in Swanson–Huneke [20, Chapters 5, 6, 14]. We also use a number of concepts considered in Lipman’s paper [16]. Let (R, m) be a regular local ring of dimension d ≥ 2. Lipman considers the structure of a certain class of complete ideals of R, the finitely supported complete ideals, in [16]. He proves a factorization theorem for the finitely supported complete ideals that extends the factorization theory of complete ideals in a two-dimensional regular local ring as developed by Zariski [22, Appendix 5]. The product of two complete ideals in a two-dimensional regular local ring is again complete. This no longer holds in higher dimension, [3] or [13]. To consider the higher dimensional case, one defines for ideals I and J the ∗-product, I ∗ J to be the completion of IJ. A complete ideal I in a commutative ring R is said to be ∗-simple if I = R and if I = J ∗ L with ideals J and L in R implies that either J = R or L = R. Another concept used by Zariski in [22] is that of the transform of an ideal; the complete transform of an ideal is used in [16] and [5]. Definition 1.1. Let R ⊆ T be unique factorization domains (UFDs) with R and T having the same field of fractions, and let I be an ideal of R not contained in any proper principal ideal. (1) The transform of I in T is the ideal I T = a−1 IT , where aT is the smallest principal ideal in T that contains IT . (2) The complete transform of I in T is the completion I T of I T . A proper ideal I in a commutative ring R is simple if I = L · H, for any proper ideals L and H. An element α ∈ R is said to be integral over I if α satisfies an equation of the form αn + r1 αn−1 + · · · + rn = 0,

where ri ∈ I i .

The set of all elements in R that are integral over an ideal I forms an ideal, denoted by I and called the integral closure of I. An ideal I is said to be complete (or, integrally closed) if I = I.

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For an ideal I of a local ring (R, m), the order of I, denoted ordR I, is r if I ⊆ mr but I  mr+1 . If (R, m) is a regular local ring, the function that associates to an element a ∈ R, the order of the principal ideal aR, defines a discrete rank-one valuation, denoted ordR on the field of fractions of R. The associated valuation ring (DVR) is called the order valuation ring of R. Let I be a nonzero ideal of a Noetherian integral domain R. The set of Rees valuation rings of I is denoted Rees I, or by ReesR I to also indicate the ring in which I is an ideal. It is by definition the set of DVRs   I  R | a Q

0 = a ∈ I

  I  and Q ∈ Spec R a

 is of height one with I ⊂ Q ,

where · denotes integral closure in the field of fractions. The corresponding discrete valuations with value group Z are called the Rees valuations of I. For any positive integer n, Rees I n = Rees I. If J ⊆ I are ideals of R and I is integral over J, then Rees J = Rees I. In this paper, we consider questions of a similar nature to the questions considered in the papers [14,15,7], in the case where a regular local ring R of dim R ≥ 3. Let (R, m, k) be a regular local ring of dimension d ≥ 2 with infinite residue field k and field of fractions Q(R), and let I be an m-primary ideal. We examine quotient rings of the Rees algebra R[It] at minimal primes of mR[It]. Let P ∈ Min(mR[It]). Then P is a homogeneous prime ideal of height one and the quotient ring R[It] is a d-dimensional P standard graded Noetherian domain over k. In Section 3, let (S, n) be a regular local ring of dim(S) = dim(R) ≥ 2 that birationally dominates R, and let V be the order valuation ring of S with corresponding valuation ν := ordS . Assume that I S = S and ν ∈ ReesS I S . Let u := αt with IS = αI S , where α ∈ S. Then V = W ∩ Q(R) with W = (R[It])Q = (S[I S u])Q , where Q ∈ Min(mR[It]) and Q ∈ Min(nS[I S u]). Let P, P  be the center of W on R[It] and S[I s u], respectively. s R In Theorem 3.7, we prove that if [ Sn : m ] = 1, then R[It] = S[IP  u] . In the case where P ordS (I S ) = 1, Corollary 3.8 shows that the quotient ring R[It] is regular and hence P polynomial ring. For the main results in Section 4, we use the following setting. Setting 1.2. Let (R, m, k) be a regular local ring of dimension d ≥ 2 with field of fractions Q(R). Let I be an m-primary finitely supported complete ideal. Let T be a terminal base point of I and let V be the mT -adic order valuation ring of T with corresponding valuation v := ordT . Let n ≥ 1 be an integer. Assume that I T = mnT . By [11, Proposition 3.11], we have v ∈ ReesR I. We observe in Remark 2.5 that V = (R[It])Q ∩ Q(R) for some Q ∈ Min(mR[It]). Let P ∈ Min(mR[It]) such that P = Q ∩ R[It]. We denote by R[It] FI (R) = mR[It] the fiber ring of an ideal I of R. R With the notation of Setting 1.2, in Section 4 we assume that [ mTT : m ] = 1. We prove R[It] ∼ in Theorem 4.3 that P = Fmn (R), which is d-dimensional normal Cohen–Macaulay

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standard graded domain over k with the multiplicity e

R[It] P N



293

= nd−1 , where N is

the unique homogeneous maximal ideal of R[It] P . In particular, we prove in Corollary 4.4 that n = 1 if and only if R[It] is a polynomial ring in d variables over k if and only if R[It] P P is regular. We show in Theorem 4.10 that k is relative algebraically closed in kv := mVV . Let (R, m, k) be a d-dimensional regular local ring with d ≥ 2. Assume that I is a ∗-product of special ∗-simple complete ideals, i.e., n n1 n2 I = PRT ∗ PRT ∗ · · · ∗ PRT , 1 2 

where n1 , . . . , n ≥ 1 are integers

where T1 , . . . , T are distinct terminal base points of I which are not equal to R. For each integer i with 1 ≤ i ≤ , let Pi := PRTi be a special ∗-simple complete ideal of R and let Vi be the order valuation ring of the terminal base point Ti and Vi = Wi ∩ Q(R) with Wi := (R[Pi t])Qi and Qi ∈ Min(mR[Pi t]). Let Pi be the center of Wi on R[It]. R In Theorem 4.5, we show that if [ mTTi : m ] = 1, then R[It] is d-dimensional normal Pi i   Cohen–Macaulay standard graded domain over k with the multiplicity e R[It] = Pi N i

nd−1 , where Ni is the unique homogeneous maximal ideal of i in Corollary 4.6 that ni = 1 if and only if

R[It] Pi

R[It] Pi .

In particular, we prove

is a polynomial ring in d variables over k

R[It] Pi

if and only if is regular. In Section 5 we prove in Theorem 5.1 that with notation as in Theorem 3.7  R[It]   S[I S u]  S R  : , = e M P N P n m

e

where N is a unique homogeneous maximal ideal of S

R[It] P ,

and M is a unique homogeneous

maximal ideal of S[IP  u] . In Theorem 5.6, we show that with notation as R [ mTT : m ] > 1, then R[It] is not Cohen–Macaulay, and hence not regular. P we show that if I T = mT , then we have [

T R : ]=1 mT m

⇐⇒

R[It] P

in Setting 1.2, if In Corollary 5.7

is regular.

We use μ(I) to denote the minimal number of generators of an ideal I, and e(R) to denote the multiplicity of a local ring (R, m), or equivalently of its maximal ideal m. We also use e(G) to denote the multiplicity of a standard graded ring G = k[G1 ] over a field k. 2. Preliminaries If R is a subring of a valuation domain V and mV is the maximal ideal of V , then the prime ideal mV ∩ R is called the center of V on R. Let (R, m) be a Noetherian local domain with field of fractions Q(R). A valuation domain (V, mV ) is said to birationally dominate R if R ⊆ V ⊆ Q(R) and mV ∩ R = m, that is, m is the center of V on R. The

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valuation domain V is said to be a prime divisor of R if V birationally dominates R and the transcendence degree of the field V /mV over R/m is dim R − 1. If V is a prime divisor of R, then V is a DVR [1, p. 330]. The quadratic dilatation or blowup of m along V , cf. [18, p. 141], is the unique local ring on the blowup Blm (R) of m that is dominated by V . The ideal mV is principal and is generated by an element of m. Let a ∈ m be such that aV = mV . Then R[m/a] ⊂ V . Let Q := mV ∩ R[m/a]. Then R[m/a]Q is the quadratic transformation of R along V . In the special case where (R, m) is a d-dimensional regular local domain we use the following terminology. Definition 2.1. Let d be a positive integer and let (R, m, k) be a d-dimensional regular local ring with maximal ideal m and residue field k. Let x ∈ m \ m2 and let S1 := R[ m x ]. The ring S1 is a d-dimensional regular ring in the sense that dim S1 = d and each localization of S1 at a prime ideal is a regular local ring. To see this, observe that S1 /xS1 is isomorphic to a polynomial ring in d − 1 variables over the field k, cf. [20, Corollary 5.5.9], and S1 [1/x] = R[1/x] is a regular ring. Moreover, S1 is a UFD since x is a prime element of S1 and S1 [1/x] = R[1/x] is a UFD, cf. [17, Theorem 20.2]. Let I be an m-primary ideal of R with r := ordR (I). Then one has in S1 IS1 = xr I1

for some ideal I1

of S1 .

It follows that either I1 = S1 or ht I1 ≥ 2. Thus I1 is the transform I S1 of I in S1 as in Definition 1.1. Let p be a prime ideal of R[ m x ] with m ⊆ p. The local ring R1 : = R[

m ]p = (S1 )p x

is called a local quadratic transform of R; the ideal I1 R1 is the transform of I in R1 as in Definition 1.1. We follow the notation of [16] and refer to regular local rings of dimension at least two as points. A point T is said to be infinitely near to a point R, in symbols, R ≺ T , if there is a finite sequence of local quadratic transformations R =: R0 ⊂ R1 ⊂ R2 ⊂ · · · ⊂ Rn = T

(n ≥ 0),

(1)

where Ri+1 is a local quadratic transform of Ri for i = 0, 1, . . . , n −1. If such a sequence of local quadratic transforms as in (1) exists, then it is unique and it is called the quadratic sequence from R to T [16, Definition 1.6]. Definition 2.2. A base point of a nonzero ideal I ⊂ R is a point T infinitely near to R such that I T = T . The set of base points of I is denoted by

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BP(I) = { T | T is a point such that R ≺ T and ordT (I T ) = 0 }. The point basis of a nonzero ideal I ⊂ R is the family of nonnegative integers B(I) = { ordT (I T ) | R ≺ T }. The nonzero ideal I is said to be finitely supported if I has only finitely many base points. The infinitely near points to R form a partially ordered set with respect to domination. The regular local ring R is the unique minimal point with respect to this partial order. For an ideal I of R, the set BP(I) of base points of I is a partially ordered subset of the set of infinitely near points to R. If the set BP(I) is finite, we refer to the maximal regular local rings in BP(I) as terminal base points of I. The set of terminal base points of I is denoted by T BP(I). Definition 2.3. Let R ≺ T be points such that dim R = dim T . Lipman proves in [16, Proposition 2.1] the existence of a unique complete ideal PRT in R such that for every point A with R ≺ A, the complete transform

(PRT

)A

is

a ∗-simple ideal if A ≺ T, the ring A otherwise.

The ideal PRT of R is said to be a special ∗-simple complete ideal. In the case where R ≺ T and dim R = dim T , we say that the order valuation ring of T is a special prime divisor of R. Remark 2.4. With notation as in Definition 2.3, a prime divisor V of R is special if and only if the unique point T with R ≺ T such that the order valuation ring of T is V has dim T = dim R. Let dim R = d. If V is a special prime divisor of R, then the residue field of V is a pure transcendental extension of degree d − 1 of the residue field T /mT of T , and T /mT is a finite algebraic extension of R/m. If the residue field R/m of R is algebraically closed and V is a special prime divisor of R, then the residue field of V is a pure transcendental extension of R/m of transcendence degree d − 1. Remark 2.5. Let (R, m, k) be a regular local ring of dimension d ≥ 1 with infinite residue field k and field of fractions Q(R), and let I be an m-primary ideal. Let R[It] denote the integral closure of the Rees algebra R[It]. Then we have (1) ([20, Exercise 10.6]) The minimal prime ideals of mR[It] are in one-to-one correspondence with the Rees valuation rings of I. The correspondence associates to each Rees valuation ring V of I a unique prime Q ∈ Min(mR[It]) such that V = R[It]Q ∩Q(R).

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(2) ([8, Remark 2.4]) For Q ∈ Min(mR[It]), the localization W = R[It]Q is a DVR and V = W ∩ Q(R) is a Rees valuation of I. Let v denote the normalized valuation associated to V . Let P ∈ Min(mR[It]). By Lying-Over and Incomparability under integral extension, there exists Q ∈ Min(mR[It]) such that P = Q ∩ R[It], and hence | Min(mR[It])| ≤ | Min(mR[It])|. By a well-known result of Rees [19], [20, Theorem 9.1.2], R[It] is a finitely generated R[It]-module. Since R is universally catenary, we have that R[It] is universally catenary, and hence Q ∩ R[It] ∈ Min(mR[It]) for each Q ∈ Min(mR[It]), by Dimension formula ([20, Theorem B.5.1]). With R and I as in Remark 2.5, if P ∈ Min(mR[It]), then P is a homogeneous prime ideal of height one and the quotient ring G := R[It] is a d-dimensional standard graded P Noetherian domain over k ([8, Remark 2.3]). It is proved in [15, Lemma 2.5] that G is a regular ring if and only if G is a polynomial ring in d-variables over the field k. In the case where dim R = 2, Huneke and Sally ([14, Theorem 3.8]) proved if I is simple complete m-primary, then R[It] is regular. In [7, Theorem 3.8] with dim R = 2, Heinzer P and Kim proved if I = K n L, where K is a simple complete m-primary ideal and either L = R, or L is an complete m-primary ideal that does not have K as a factor, then R[It] P is a two dimensional normal Cohen–Macaulay standard graded domain over k that has minimal  multiplicity at its maximal homogeneous ideal N with this multiplicity being N e R[It] in the case where dim R ≥ 3. = n. We consider the structure of R[It] PN P 3. The transform of an ideal Setting 3.1. Let (R, m, k) and (S, n) be regular local rings having the same dimension d ≥ 2 and S birationally dominates R. Let I be an m-primary ideal of R such that the transform of I in S, I S = S. Let V be the order valuation ring of S with corresponding valuation ν := ordS . Assume that ν ∈ ReesS I S . By [11, Proposition 3.11], we have ν ∈ ReesR I. We observe in Remark 2.5 that V = (R[It])Q ∩Q(R) for some Q ∈ Min(mR[It]). Let P ∈ Min(mR[It]) be such that P = Q ∩ R[It]. Remark 3.2. Let the notation be as in Setting 3.1. Since the transform I S = S, there exists nonzero element α ∈ S such that IS = αI S . Hence we have R[It] ⊂ R[It] ⊂ S[I S u],

where u := αt .

Since ν ∈ ReesS I S , by [11, Proposition 3.11], we have ν ∈ ReesR I. Hence we have W = (R[It])Q , = (S[I S u])Q ,

Q ∈ Min(mR[It]) Q ∈ Min(nS[I S u]),

with V = W ∩ Q(R).

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That is, we have R[It]

S[I S u]

W

(2)

g

f

R[It]

S[I S u]

where f1 and f2 are inclusion maps. Let P  := Q ∩ S[I S u]. Then P  ∈ Min(nS[I S u]). Notice that Q ∩ R[It] = P and we have the following inclusion: S[I S u] R[It] W ⊂ ⊂  P P mW Then R[It] and P respectively.

S[I S u] P

are d-dimensional standard graded domain over field

R m

and

S n,

Notation 3.3. Let (R, m, k) be a Noetherian local domain, let V be a DVR that birationally dominates R and let v be the corresponding valuation with value group Z. Let kv := mVV denote the residue field of V . For a nonzero ideal L of R, we define LV := {a ∈ L | aV = LV }. Then we have: (1) LV ⊆ L and v(mL) = v(m) + v(L) > v(L). Hence mL ⊆ LV and thus LLV is an R/m =: k-vector space. (2) Since there exists an element  ∈ L such that V = LV , we always have  dimk LLV ≥ 1. Notice that if L := R, then LV = m.  (3) If d := dimk LLV , then there exist elements a1 , . . . , ad , c1 , . . . , cm in L such that {a1 , . . . , ad , c1 , . . . , cm } is a minimal generating set for L and the images a1 , . . . , ad in LLV form a basis of LLV over k. Remark 3.4. Let the notation be as in Setting 3.1 and Remark 3.2. Then we have R[It]  R  I  = and P m IV I   R[It]  = dim R and edim m P IV

S[I S u]  S  I S  = P n (I S )V  S[I S u]   IS  S edim = dim . n P (I S )V

We shall show in Theorem 3.7 that if there is no residue extension from R to S as in S Setting 3.1 and Remark 3.2, then R[It] = S[IP  u] . To see this, we use Lemma 3.5. P

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Lemma 3.5. Let the notation be as in Setting 3.1 and Remark 3.2. Assume that R [ Sn : m ] = 1. Then we have dimk

I   IS  = dimk . IV (I S )V

R R Proof. Let k := m = Sn , by assumption [ Sn : m ] = 1. Let τ := dimk μ := μ(I). Then there exist the following elements in I



I IV

 and let

ξ1 , ξ2 , . . . , ξτ , ξτ +1 , . . . , ξμ such that {ξ1 , . . . , ξτ , ξτ +1 , . . . , ξμ } is a minimal generating set of I and ν(ξk ) > ν(ξ1 ) = · · · = ν(ξτ ) = ν(I) for k with τ + 1 ≤ k ≤ μ, and the images ξ1 , ξ2 , . . . , ξτ in IIV form a basis of IIV over k. Since IS = αI S , we have η1 :=

ξ1 ξτ , . . . , ητ := , α α

ητ +1 :=

ξτ +1 ξμ . . . , ημ := α α

generate I S . Since ν(ηk ) = ν(ξk ) − ν(α) > ν(ξ1 ) − ν(α) = ν(η1 ) = ν(I) − ν(α) = ν(I S ) (I S )V , andhencethe images for k with τ + 1 ≤ k ≤ μ, we  havethat ητ +1 , . . . , ημ are in  S

η1 , η2 , . . . , ητ generate (IIS )V over k. Hence dimk that equality holds, we want to claim the following. Claim 3.6. The images η1 , . . . , ητ in

IS (I S )V

I IV

≥ dimk

IS (I S )V

. To show

are linearly independent over k.

Proof of Claim. Suppose that c1 η1 +· · ·+cτ ητ = 0 in Then we have

IS (I S )V

, where ci ∈ k for i = 1, . . . , τ .

c1 η1 + · · · + cτ ητ ∈ (I S )V ⇐⇒ ν(c1 η1 + · · · + cτ ητ ) > ν(I S ) ⇐⇒ ν(c1 ξ1 + · · · + cτ ξτ ) − ν(α) > ν(I S ) ⇐⇒ ν(c1 ξ1 + · · · + cτ ξτ ) > ν(I S ) + ν(α) = ν(I) ⇐⇒ c1 ξ1 + · · · + cτ ξτ ∈ IV ⇐⇒ c1 ξ1 + · · · + cτ ξτ = 0 Since the images ξ1 , . . . , ξτ in IIV form a basis of This completes the proof of Claim. 2 This completes the proof of Lemma 3.5. 2

I IV

in

I . IV

over k, we have that c1 = · · · = cτ = 0.

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Theorem 3.7. Let the notation be as in Setting 3.1 and Remark 3.2. Assume that R [ Sn : m ] = 1. Then we have S[I S u] R[It] = . P P Proof. From Remark 3.2, we have the following inclusion: S[I S u] R[It] ⊂ , P P R and both are d-dimensional standard graded domains over m and Sn , respectively. Since R S R S [ m : n ] = 1, we have k := m = n . From Remark 3.4 and Lemma 3.5, we have

edim Hence we have

 R[It]  P

R[It] P

=

= dimk

S[I S u] P .

I   IS   S[I S u]  = dimk = edim . IV (I S )V P 2

Corollary 3.8. Let the notation be as in Setting 3.1 and Remark 3.2. Assume that R [ Sn : m ] = 1. If ordS (I S ) = 1, then R[It] is regular, and hence polynomial ring. P Proof. Notice that ν := ordS is a prime divisor of S that is the only Rees valuation of n, (cf. [20, Example 10.3.1]). Since ν ∈ ReesS I S , we may assume that I S is n-primary, by [20, Proposition 10.4.4]. Since ν ∈ ReesS I S and ordS (I S ) = 1, we have I S = n by [11, Lemma 3.13]. From Theorem 3.7, we have S[I S u] R[It] S[nu] = = grn (S) = Fn (S) ∼ = = k[X1 , X2 , . . . , Xd ], P P nS[nu] where X1 , X2 , . . . , Xd are variables over k. Hence ring. 2

R[It] P

is regular, and hence polynomial

Lipman gives the following example to illustrate the decomposition as in [16, Theorem 2.5]. Example 3.9 illustrates a situation where v0 := ordR0 ∈ ReesR I and R[It] P0 is regular. However, ordR (I) > 1. Example 3.9. Let k be a field and let R0 := R = k[[x, y, z]] be the formal power series ring in the 3 variables x, y, z over k, and let m := (x, y, z)R. Let x

R1 := R

m x

y z (x, x ,x)

,

y

R1 := R

m y

z (x y ,y, y )

,

z

R1 := R

m z

y (x z , z ,z)

be the local quadratic transformations of R in x, y, z directions. The associated special ∗-simple complete ideals are

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300

PR0 x R1 = (x2 , y, z)R,

PR0y R1 = (x, y 2 , z)R,

PR0z R1 = (x, y, z 2 )R.

The equation m(x3 , y 3 , z 3 , xy, xz, yz) = PR0 x R1 PR0y R1 PR0z R1 represents the factorization of the finitely supported ideal I = (x3 , y 3 , z 3 , xy, xz, yz)R as a product of special ∗-simple complete ideals. Here PR0 R0 = (x, y, z)R. Then we have (1) BP(I) = {R0 , x R1 , y R1 , z R1 } and T BP(I) = {x R1 , y R1 , z R1 }. (2) ReesR I = {v0 := ordR0 , v1 := ordx R1 , v2 := ordy R1 , v3 := ordz R1 }. (3) Let Pi be the center of Wi := (R[It])Qi on R[It], where Qi ∈ Min(mR[It]) for i = 0, 1, 2, 3. x y z (a) Since ordx R1 (I R1 ) = ordy R1 (I R1 ) = ordz R1 (I R1 ) = 1, we have that R[It] Pi is regular, by Corollary 3.8. (b) The quotient ring R[It] P0 = (R/m)[xyt, xzt, yzt], which is polynomial ring, and hence regular. However, ordR0 (I) = 2 > 1. 4. The normal and Cohen–Macaulay property of

R[It] P

Setting 4.1. Let (R, m, k) be a regular local ring of dimension d ≥ 2 with field of fractions Q(R). Let I be an m-primary finitely supported complete ideal. Let T ∈ T BP(I). Let V be the mT -adic order valuation ring of T with corresponding valuation v := ordT . Then there exists a unique sequence of local quadratic transformation of R along V R =: R0 ⊂ R1 ⊂ · · · ⊂ Rs := T ⊂ V

s ≥ 0,

(3)

where (Ri+1 , mi+1 ) is a local quadratic transformation of (Ri , mi ) for i = 0, 1, . . . , s − 1. Since dim(Ri ) = dim(R), we have the following. Let αi ∈ mi \ m2i Ri+1 := Ri

m  i

αi

Mi

,

  i where Mi is any maximal ideal of Ri m containing mi = (αi1 , . . . , αid )Ri for i with αi 0 ≤ i ≤ s−1. Let I0 := I and let Ii+1 be the transform of Ii in Ri+1 and let ri := ordRi (Ii ) for i with 0 ≤ i ≤ s − 1. Then one has in Ri+1 Ri+1

Ii Ri+1 = αiri Ii

Ri+1

for some ideal Ii+1 := Ii

= Ri+1 .

Let n ≥ 1 be an integer. Assume that I T = mnT . By [11, Proposition 3.11], we have v ∈ ReesR I. We observe in Remark 2.5 that V = (R[It])Q ∩Q(R) for some Q ∈ Min(mR[It]). Remark 4.2. Let (R, m, k) be a regular local ring of dim(R) = d ≥ 2. Then the fiber cone of m is the ring

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Fm (R) =

301

mi m R mi ⊕ = ⊕ · · · ⊕ ⊕ ··· ∼ = k[X1 , X2 , . . . , Xd ], mi+1 m m2 mi+1 i≥0

i

where X1 , X2 , . . . , Xd are variables over a field k. Let Fi := mmi+1 be the degree i com ponent of Fm (R) for i ≥ 0. Then Fm (R) = i≥0 Fi . Let n ≥ 1 be an integer. Then the n-th Veronese subring of Fm (R) is Fmn (R) =

i≥0

Fni =

mn R m2n R[mn t] ⊕ n+1 ⊕ 2n+1 ⊕ · · · = . m m m mR[mn t]

Let Mn denote the set of monomials of degree n in k[X1 , X2 , . . . , Xd ]. That is, Mn = {X1λ1 X2λ2 · · · Xdλd | λ1 + · · · + λd = n,

0 ≤ λi ≤ n,

1 ≤ i ≤ d}.

 n Then Mn forms vector space over k with dimk (Mn ) = d+n−1 d−1 . Hence Fm (R) = k[Mn ] is normal, (cf. [2, Theorem 6.1.4]). By [12, Theorem 1], it follows that Fmn (R) is Cohen–Macaulay. That is, Fmn (R) is a d-dimensional normal Cohen–Macaulay stan dard graded domain over k. Moreover, the multiplicity e (Fmn (R))M = nd−1 , (cf. [21, Exercise 18.6.J]), where M is unique homogeneous maximal ideal in Fmn (R). Theorem 4.3. Let the notation be as in Setting 4.1. Let n ≥ 1 be an integer. Assume R that I T = mnT and [ mTT : m ] = 1. Let P ∈ Min(mR[It]) be such that P = Q ∩ R[It] and V = (R[It])Q ∩ Q(R) where Q ∈ Min(mR[It]). Then we have R[It] ∼ = Fmn (R), P which is d-dimensional normal Cohen–Macaulay standard graded domain over k with   R[It] d−1 the multiplicity e = n , where N is the unique homogeneous maximal ideal P N of

R[It] P .

Proof. Let Rs := T in Equation (3). Notice that ReesRs I Rs = ReesT mnT = {ordT }. By [11, Proposition 3.11], we have v := ordT ∈ ReesRi Ii , for all i = 0, 1, 2, . . . , s. Hence we have V = Wi ∩ Q(R0 )

and Wi = (Ri [Ii t])Qi ,

where Qi ∈ Min(mi Ri [Ii t]) Ri+1

for i with 0 ≤ i ≤ s. Let W := W0 . Since Ii Ri+1 = αiri Ii t0 := t

and ti+1 := αiri ti

, we let

for i with 0 ≤ i ≤ s − 1.

Then we have the following inclusions of graded rings:

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302

R0 [I0 t0 ]

R1 [I1 t1 ]

f0

R2 [I2 t2 ]

f1

R0 [I0 t0 ]

···

f2

R1 [I1 t1 ]

R2 [I2 t2 ]

Rs [mns ts ]

W

fs

···

Rs [mns ts ]

where f0 , f1 , . . . , fs are inclusion maps. Notice that W = (Ri [Ii ti ])Qi where Qi ∈ Min(mi Ri [Ii ti ]) for i with 0 ≤ i ≤ s. Notice that Q = Q0 = Q0 and Qs = Qs = ms Rs [mns ts ]. For i with 0 ≤ i ≤ s, let Pi := Qi ∩ Ri [Ii ti ]

: the center of W on Ri [Ii ti ].

Notice that P = P0 = P0 , and Ps = Qs = ms Rs [mns ts ], since mns is normal in Rs . Hence we have the following inclusions of graded rings: G :=

R0 [I0 t0 ] R[It] R1 [I1 t1 ] R2 [I2 t2 ] Rs [mns ts ] W = ⊆ ⊆ ⊆ ··· ⊆ =: B ⊆ .    P P0 P1 P2 Ps mW (4)

By assumption Is = mns , we have the following by Lemma 3.5 dimk

 I   I   I   mn  1 s s = dimk = · · · = dimk = dimk IV (I1 )V (Is )V mn+1 s

By Theorem 3.7, we have R0 [I0 t0 ] R[It] R1 [I1 t1 ] R2 [I2 t2 ] Rs [mns ts ] =: = = = · · · = . P P0 P1 P2 Ps By Remark 4.2, we have B=

Rs [mns ts ] = Fmns (Rs ) ∼ = Fmn (R) = k[Mn ], ms Rs [mns ts ]

which is d-dimensional normal Cohen–Macaulay domain with the multiplicity e(BM ) = ∼ nd−1 , where M is unique homogeneous maximal ideal in B. Hence, we have R[It] = P n Fm (R), which is d-dimensional normal Cohen–Macaulay standard graded domain over k with the multiplicity nd−1 . 2 We recored in Corollary 4.4 the case where the exponent of mT is equal to one. Corollary 4.4. Let the notation be as in Setting 4.1. Let n ≥ 1 be an integer. Assume R that I T = mnT and [ mTT : m ] = 1. Let P ∈ Min(mR[It]) be such that P = Q ∩ R[It] and V = (R[It])Q ∩ Q(R) where Q ∈ Min(mR[It]). Then the following are equivalent:

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303

(1) n = 1. (2) R[It] is a polynomial ring in d variables over k. P (3)

R[It] P

is regular.

Proof. (1) ⇒ (2): It follows from Theorem 4.3. (2) ⇒ (1): From the proof of Theorem 4.3, we have R0 [I0 t0 ] R1 [I1 t1 ] R2 [I2 t2 ] Rs [mns ts ] R[It] =: = = = ··· = =: B.   P P0 P1 P2 Ps By Remark 4.2, we have B=

Rs [mns ts ] = Fmns (Rs ) ∼ = Fmn (R) = k[Mn ]. ms Rs [mns ts ]

Since R[It] is a polynomial ring in d variables over k, dimk (Mn ) = P n = 1. (2) ⇐⇒ (3): It is proved in [15, Lemma 2.5]. 2

d+n−1 d−1

= d. Hence

The following is generalization of result of Heinzer and Kim ([7, Theorem 3.8]) in the case where dim(R) ≥ 3. Theorem 4.5. Let (R, m, k) be a d-dimensional regular local ring with d ≥ 2. Let I = P1n1 ∗ P2n2 ∗ · · · ∗ Pn ,

where n1 , . . . , n ≥ 1 are integers

where the products denotes ∗-products and where P1 , P2 , . . . , P are special ∗-simple complete ideals which are not equal to m, and are associated with distinct terminal base points T1 , . . . , T , (i.e., Pi = PRTi for i with 1 ≤ i ≤ ). For each integer i with 1 ≤ i ≤ , let Vi be the order valuation ring of the terminal base point Ti and Vi = Wi ∩ Q(R) with Wi := (R[Pi t])Qi where Qi ∈ Min(mR[Pi t]). Let Pi be the center of Wi on R[It]. R Assume that [ mTTi : m ] = 1 for i with 1 ≤ i ≤ . Then R[It] is d-dimensional normal Pi i   Cohen–Macaulay standard graded domain over k with the multiplicity e R[It] = Pi Ni , where Ni is the unique homogeneous maximal ideal of nd−1 i

R[It] Pi .

Proof. Let R := P1n1 P2n2 · · · Pn denote the usual products. Notice that R is a finitely supported ideal in R. By definition of ∗-products, we have I = R , i.e., I = P1n1 P2n2 · · · Pn ,

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and hence R ⊆ I = R . Since T BP(Pi ) = {Ti }, we have T BP(I) = {T1 , . . . , T }. For i with 1 ≤ i ≤ , we have by [16, Proposition 1.5], ni Ti Ti ni (R)Ti ⊆ I Ti ⊆ RTi = (Pini )Ti = (PRT ) = (PRT ) = mnTii . i i

Hence we have mnTii ⊆ I Ti ⊆ mnTii = mnTii , and thus I Ti = mnTii . By Theorem 4.3, we have R[It] ∼ = Fmni (R) is d-dimensional normal Cohen–Macaulay standard graded domain Pi   over k with the multiplicity e R[It] , where Ni is the unique homogeneous = nd−1 i Pi N i

maximal ideal of

R[It] Pi .

2

As an immediate consequence of Theorem 4.5 and Corollary 4.4 we have the following. Corollary 4.6. Let notation be as in Theorem 4.5. For each i with 1 ≤ i ≤ , assume that R [ mTTi : m ] = 1. Then the following are equivalent: i

(1) ni = 1. (2) R[It] Pi is a polynomial ring in d variables over k. (3)

R[It] Pi

is regular.

Example 4.7 demonstrates that the equivalent statement of items (1) and (2) in Corollary 4.4 need not be true if I is not finitely supported. Example 4.7. Let (R, m, k) be a 3-dimensional regular local ring with residue field R/m = k and maximal ideal m := (x, y, z)R. Let I := (x2 , xy, y 2 , z)R

and S1 := R

m x

and x1 := x, y1 :=

y z , z1 := . x x

Then we have: (1) I is an m-primary complete ideal of R. (2) Since J := (x2 , y 2 , z)R is a reduction of I, we have ReesR I = ReesJ I = {v}, where v(x) = 1, v(y) = 1, and v(z) = 2. (3) p := I S1 = (x1 , z1 )S1 is a prime ideal in S1 with ht(I S1 ) = 2. Let A1 := (S1 )p . Then (A1 , pA1 ) is a 2-dimensional regular local ring which is a local quadratic transform of R, and I A1 = pA1 =: mA1 . (4) I is not finitely supported, since dim(R) = dim(A1 ). (5) A1 ∈ T BP(I), since I A1 = mA1 . (6) ReesR I = ReesS1 I S1 = ReesA1 I A1 = {ordA1 }, (cf. [10, Example 4.10]). (7) Let V be the order valuation ring of A1 with corresponding valuation v = ordA1 . By Remark 2.5, we have V = (R[It])Q ∩ Q(R) and {Q} = Min(mR[It]). Notice that Min(mR[It]) = {P }, where P = mR[It] = Q ∩ R[It]. Let ∗ denote the image of ∗

M.-K. Kim / Journal of Algebra 488 (2017) 290–314

in

R[It] P .

305

Then we have R[It] = k[x2 t, xyt, y 2 t, zt]. P

(8) Let X1 , X2 , X3 , X4 be variables over k and consider the following k-algebra map φ : k[X1 , X2 , X3 , X4 ] −→

R[It] = k[x2 t, xyt, y 2 t, zt] P

given by φ(X1 ) = x2 t,

φ(X2 ) = xyt,

R[It] ∼ k[X1 ,X2 ,X3 ,X4 ] = (X1 X3 −X 2 ) P 2 R[It] However, P is not regular.

Hence

φ(X3 ) = y 2 t,

φ(X4 ) = zt.

is a 3-dimensional normal Cohen–Macaulay domain.

We present in Example 4.8 a specific example where the hypotheses of Theorem 4.3 R[It] hold. Moreover, Example 4.8 illustrates that R[It] P0 and P1 are 3-dimensional normal Cohen–Macaulay standard graded domain over k, where for i = 0, 1, Pi is the center of Wi := (R[It])Qi on R[It] and Vi := Wi ∩ Q(R) denotes the valuation ring corresponding to vi := ordRi , and {R0 , R1 } = BP(I) \ T BP(I). Example 4.8. Let k be a field and let R0 := R = k[[x, y, z]] be the formal power series ring in the 3 variables x, y, z over k, and let m := (x, y, z)R. Let R1 := x R1 := R

m x

y z and m1 := (x1 , y1 , z1 )R1 = (x, , )R1 x x

y z (x, x ,x)

denote the local quadratic transformations of R in x direction. Let R2 := yx R2 := R1

m 

and R2 := zx R2 := R1

1

y1

x 1

z

( y1 ,y1 , y1 ) 1

m  1

z1

x 1

y 1

( z 1 , z1 ,z1 )

denote the local quadratic transformations of R1 in y, z directions, and mR2 := (

x1 z1 , y1 , )R2 y1 y1

and mR2 := (

x1 y1 , , z1 )R2 . z1 z1

The associated special ∗-simple ideals are PR0 R2 = (x3 , y 2 , xz, x2 y, yz, z 2 )R

and PR0 R2 = (x3 , z 2 , xy, x2 z, zy, y 2 )R.

The equation  mI =

2 PR0 R2

PR0 R2

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306

represents the factorization of the finitely supported complete ideal  I = x8 , x7 z, x6 y, x6 z, x5 y 2 , x4 yz, x4 z 2 , x3 y 3 , x3 y 2 z, x3 z 3 ,  x2 y 4 , x2 yz 2 , xy 3 z, xy 2 z 2 , xyz 3 , xz 4 , (y, z)5 R0 as a product of special ∗-simple ideals. Then we have (1) BP(I) = {R0 , R1 , R2 , R2 } and T BP(I) = {R2 , R2 }.  2 (2) I R1 = PR1 R2 PR1 R2 = (x3 , x2 y1 , x2 z1 , xy13 , xy1 z1 , xz12 , y15 , y13 z1 , y1 z12 , z14 )R1 . 

(3) I R2 = (mR2 )2 and I R2 = mR2 . (4) ReesR I = {v0 := ordR0 , v1 := ordR1 , v2 := ordR2 , v2 := ordR2 }. (5) Let V2 denote the valuation ring of R2 corresponding to v2 := ordR2 , where V2 = W2 ∩ Q(R). Let V2 denote the valuation ring of R2 corresponding to v2 := ordR2 , where V2 = W2 ∩ Q(R). Let P2 be the center of W2 := (R[It])Q2 on R[It] and let P2 be the center of W2 := (R[It])Q2 on R[It], where Q2 , Q2 ∈ Min(mR[It]). Let X, Y, Z be indeterminate over k. Then we have 2 2 2 ∼ (a) R[It] P2 = k[X , XY, XZ, Y , Y Z, Z ] is a 3-dimensional normal Cohen–Macaulay standard graded domain over k that has minimal multiplicity at its maximal  N 2 homogeneous ideal N with this multiplicity being e R[It] = 2 = 4. P2N R[It] ∼ (b) = k[X, Y, Z] is a polynomial ring, and hence regular.  P2

(6) For i = 0, 1, let Vi denote the valuation ring of Ri corresponding to vi := ordRi , where Vi = Wi ∩ Q(R) and Wi := (R[It])Qi and Qi ∈ Min(mR[It]). Let Pi be the center of Wi on R[It]. Then we have (a) R[It] P1 is a 3-dimensional normal Cohen–Macaulay standard graded domain over k. (b)

R[It] P0

is a 3-dimensional normal Cohen–Macaulay standard graded domain over k.

Proof. Items (1), (2), and (3) follow from direct computations. (4): The valuations are defined by

v2 v2 v1 v0

:= := := :=

ordR2 ordR2 ordR1 ordR0

x

y

z

2 2 1 1

3 4 2 1

4 3 2 1

We have the following table: I

x8

x7 z

x6 y

x6 z

x5 y 2

x4 yz

x4 z 2

x3 y 3

x3 y 2 z

x3 z 3

x2 y 4

x2 yz 2

v2 v2 v1 v0

16 16 8 8

18 17 9 8

15 16 8 7

16 15 8 7

16 18 9 7

15 15 8 6

16 14 8 6

15 18 9 6

16 17 9 6

18 15 9 6

16 20 10 6

15 14 8 5

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I

xy 3 z

xy 2 z 2

xyz 3

xz 4

(y, z)5 = y 5 , y 4 z, · · · , yz 4 , z 5

v2 v2 v1 v0

15 17 9 5

16 16 9 5

17 15 9 5

18 14 9 5

15, 16, 17, · · · , 19, 20 20, 19, · · · , 16, 15 10, 10, · · · , 10, 10 5, 5, · · · , 5, 5

307

Notice that x3 y 3 xy 2 xy 3 z y2 = 2 , = 2 2 2 2 x yz z x yz xz are algebraically independent over k,

the images of in kv2

x4 z 2 xz 4 x2 z2 , = = 2 2 2 2 x yz y x yz xy are algebraically independent over k,

the images of in kv2

x4 z 2 x4 yz x2 x2 , = = 2 2 2 2 x yz z x yz y are algebraically independent over k,

the images of in kv1

2 2

and

3

xyz xy z y z = , = 2 2 2 2 x yz x x yz x are algebraically independent over k.

the images of in kv0

By [11, Theorem 3.9], we have {v2 , v2 , v1 , v0 } ⊆ ReesR I. Hence by [11, Proposition 3.11], we have ReesR I = {v0 , v1 , v2 , v2 }. ∼ (5)-(a): By item (3) and Theorem 4.3, R[It] = k[X 2 , XY, XZ, Y 2 , Y Z, Z 2 ] is a P2 2-dimensional normal Cohen–Macaulay standard graded domain over k with the mul  R[It]N tiplicity e P2 = 22 = 4, where N is unique homogeneous maximal ideal of R[It]. N      N Hence e R[It] = edim R[It] − dim R[It] + 1 = 6 − 3 + 1 = 4, and thus the ring P2 P2 P2 N

R[It] P2

has minimal multiplicity.

∼ (b): By item (3) and Corollary 4.4, R[It] P2 = k[X, Y, Z] is a polynomial ring, and hence regular. (6)-(a): Notice that ordR (I) = 5. Let t1 := x5 t. Let P1 be the center of W1 on R1 [I R1 t1 ]. Since v1 := ordR1 ∈ ReesR1 I R1 , we have by Theorem 3.7 R[It] R1 [I R1 t1 ] W1 = ⊂ . P1 P1 m W1 ∗ denote the image of ∗ in Let 

I R1 (I R1 )V1

. Then we have

 I R1  R1 [I R1 t1 ] = k P1 (I R1 )V1   2 , yz 2 ] 3 , x 2 y, x 2 z, xyz,  xz = k[x = k[X 3 , X 2 Y , X 2 Z, XY Z, XZ 2 , Y Z 2 ],

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308

where X, Y, Z denote the image of x, y, z in on k[X, Y, Z] by Y > X > Z, we have

R1 [I R1 t1 ] . P1

With the lexicographical ordering

Y 1X 2Z 0 > Y 1X 1Z 1 > Y 1X 0Z 2 > Y 0X 3Z 0 > Y 0X 2Z 1 > Y 0X 1Z 2 Hence we have k[X 3 , X 2 Y , X 2 Z, XY Z, XZ 2 , Y Z 2 ] =: LY X 2 , generated by monomials in the lex-segment set

XZ 2

is a k-subalgebra

L(Y X 2 , XZ 2 ) := {α ∈ M3 | Y X 2 ≥ α ≥ XZ 2 }, where M3 denotes the set of monomials of degree 3 in k[X, Y, Z]. By [4, Proposition 2.14], we have k[X 3 , X 2 Y , X 2 Z, XY Z, XZ 2 , Y Z 2 ] =: LY X 2 , XZ 2 is normal, and hence Cohen–Macaulay. ∗ denote the image of ∗ in IVI . We have (6)-(b): Let  0

 I  R[It] 4 , y5 , y 4 , z5 ] 2 yz 2 , xy 3 z, xy 2 z 2 , xyz 4 z, y 3 z 2 , y 2 z 3 , yz   3 , xz =k = k[x P0 IV0 = k[X 2 Y Z 2 , XY 3 Z, XY 2 Z 2 , XY Z 3 , XZ 4 , Y 5 , Y 4 Z, Y 3 Z 2 , Y 2Z 3, Y Z 4, Z 5] R[It] where X, Y, Z denote the image of x, y, z in R[It] P0 . We check that P0 is normal using Macaulay2 ([6]), and hence Cohen–Macaulay by [12, Theorem 1]. 2

Remark 4.9. Let notation be as in Setting 4.1. With respect to the canonical embedding kv of k → kv := mVV , let k denote the set of all elements in kv that are algebraic over k. kv

The field k is said to be relatively algebraically closed in kv , if k = k . Theorem 4.10. Let notation be as in Setting 4.1. Let n ≥ 1 be an integer. Assume that R I T = mnT and [ mTT : m ] = 1. Then k is relatively algebraically closed in kv . Proof. Since V is the mT -adic order valuation ring of T with corresponding valuation v := ordT , we have v ∈ ReesRi Ii for i with 0 ≤ i ≤ s, by [11, Proposition 3.11].  Hence V = W ∩ Q(R)  and W := W0 = (R0 [I0 t0 ])Q0 where Q0 = Q0 ∈ Min(mR0 [I0 t0 ]). Then mWW = mVV (at) and W = V (at), where a ∈ I =: I0 with aV = IV . Since P0 = Q0 ∩ R[I0 t0 ], we have Rs [mns ts ] W R0 [I0 t0 ] R0 [I0 t0 ] Rs [mns ts ] ⊂ ⊂ ··· ⊂ = ⊂ .   P0 Q0 Qs Ps mW By Theorem 4.3, we have G :=

R0 [I0 t0 ] Rs [mns ts ] R0 [I0 t0 ] Rs [mns ts ] = = ··· = = ,   P0 Q0 Qs Ps

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309

which is d-dimensional normal Cohen–Macaulay standard graded domain over k and field of fractions Q(G) = mWW =: kw . Let α ∈ kv := mVV be algebraic over k. By [9, kw

Proposition 2.5], G0 = k is relatively algebraically closed in Q(G), (i.e., k = k ). Since kv ⊂ kw = (kv )(at), we have α ∈ k. Thus k is relatively algebraically closed in kv . 2 As an immediate consequence of Theorem 4.10 and Theorem 4.3 we have the following. Corollary 4.11. Let notation be as in Setting 4.1. Let n ≥ 1 be an integer. Assume that I T = mnT . Let P ∈ Min(mR[It]) be such that P = Q ∩ R[It] and V = (R[It])Q ∩ Q(R) where Q ∈ Min(mR[It]). Then the following statements are equivalent. (1) k is relatively algebraically closed in kv . (2) R[It] is d-dimensional normal standard graded domain over k. P 5. The multiplicity of

R[It] P

With notation in Setting 3.1, we prove in Theorem 5.1 that  R[It]   S[I S u]  S R  : = e , M P N P n n

e

where N is a unique homogeneous maximal ideal of maximal ideal of

S

R[It] P ,

and M is a unique homogeneous

S[I u] P .

Theorem 5.1. Let notation be as in Setting 3.1 and Remark 3.2. We have A :=

R R[It] φ S[I S u]  S  W

→ = , [A1 ] = [D1 ] =: D ⊂ m P P n mW

where the map φ is a graded inclusion of standard graded domains, and mWW is the field of fractions of both A and D. Let N be a unique homogeneous maximal ideal of A, and let M be a unique homogeneous maximal ideal of D. Then we have (1) The map φ : A → D is a graded integral birational extension of d-dimensional standard graded domains.   R (2) The multiplicity e(AN ) of A is e(DM ) Sn : m . That is, we have  R[It] 

e

P

N

 S[I S u]  S R  : . M P n n

=e

R Proof. Item 1: Since Sn := L is a finite algebraic field extension of m := k, the degree zero component D0 = L of D is a finite integral extension of the degree zero component A0 = k of A.

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310

Claim 5.2. LA1 = D1 . Proof of Claim. “⊆” is clear. “⊇”: Let β ∈ I S \ (I S )V . Then we have (β)u = (β)αt ∈ It,   and ν(βα) = ν(β) + ν(α) = ν(I S ) + ν(I) − ν(I S ) = ν(I). Hence we have βα ∈ I \ IV . This completes the proof of Claim 5.2. 2 Claim 5.3. A1 D = D1 D = M . Proof of Claim. A1 D = A1 (L ⊕ D1 ⊕ D2 ⊕ D3 ⊕ · · · ⊕ Dj ⊕ · · · ) = 0 ⊕ LA1 ⊕ A1 D1 ⊕ A1 D2 ⊕ · · · ⊕ A1 Dj ⊕ · · · = 0 ⊕ D1 ⊕ (D1 )2 ⊕ D1 D2 ⊕ · · · ⊕ D1 Dj1 ⊕ · · · = 0 ⊕ D1 ⊕ D2 ⊕ D3 ⊕ · · · ⊕ Dj+1 ⊕ · · · = D1 D = M, since LA1 = D1 by Claim 5.2, and Dj = (D1 )j for j ≥ 1. 2 Claim 5.4. D is a finite A-module. Proof of Claim. D = L ⊕ D+ = L ⊕ A1 D, and hence D is a finite A-module. 2 Item 2: Let N be a unique homogeneous maximal ideal of A. By [22, Corollary 1, p. 299] or [20, Theorem 11.2.7], we have D

A M N S R S R : = e(DM ) : 2 = e(M DM ) n m n m

e(AN ) = e(A1 AN ) = e(A1 DM )

:

Corollary 5.5 follows from consequences of Theorem 5.1 and Theorem 4.3. Corollary 5.5. Let notation be as in Setting 4.1. Let n ≥ 1 be an integer. Assume I T = mnT . Let P ∈ Min(mR[It]) be such that P = Q ∩ R[It] and V = W ∩ Q(R) with W := (R[It])Q where Q ∈ Min(mR[It]). From Equation (4), we have G :=

φ R0 [I0 t0 ]  R0  R[It] Rs [mns ts ]  Rs  W := = = , [G1 ] → B := [B1 ] ⊂ P P0 m0 Ps ms mW

where Rs := T in Equation (3) and the map φ is a graded inclusion of standard graded domains, and mWW is the field of fractions of both G and B. Let N be a unique homogeneous maximal ideal of G, and let M be a unique homogeneous maximal ideal of B. Then we have

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311

(1) The map φ : G → B is a graded integral birational extension of d-dimensional standard graded domains.   Rs R0 (2) The multiplicity e(GN ) of G is e(BM ) m : m0 . That is, we have s  R[It]   R [mn t ]  R R0  s s s s : = e .  M P N Ps ms m0

e

R Theorem 5.6. Let the notation be as in Setting 4.1. Assume that [ mTT : m ] > 1 and T I = mT . Let P ∈ Min(mR[It]) be such that P = Q ∩ R[It] and V = (R[It])Q ∩ Q(R) where Q ∈ Min(mR[It]). Then the quotient ring G := R[It] is not Cohen–Macaulay, and P hence not regular. s [ms ts ] Proof. Let Rs := T in Equation (3). Since I Rs = ms , we have B = mRs R = s [ms ts ] R Rs s ∼ Fms (Rs ) = ms [X1 , X2 , . . . , Xd ], where X1 , X2 , . . . , Xd are variables over ms . Hence the   Rs R multiplicity of B, e(BM ) = 1. By Corollary 5.5, we have e(GN ) = e(BM ) m : m = s   Rs R ms : m =: τ . Suppose that G is Cohen–Macaulay. Then by [20, Exercise 11.10, p. 233], we have

e(GN ) ≥ edim(GN ) − dim(GN ) + 1. However,  I   I  s = dimk by Lemma 3.5 IV (Is )V  I  R  I  R s s s dimks : = = dimk (ks ) dimks (Is )V ms m (Is )V R R s = : μ(ms ) ms m = τ d.

edim(GN ) = dimk

Hence we have   edim(GN ) − dim(GN ) + 1 − e(GN ) = (τ d − d + 1) − τ = (d − 1)(τ − 1) > 0 which is a contradiction. Hence GN is not a Cohen–Macaulay local ring, and therefore G = R[It] is not Cohen–Macaulay, and hence not regular. 2 P As an immediate consequence of Corollary 4.4 and Theorem 5.6 we have the following. Corollary 5.7. Let the notation be as in Setting 4.1. Assume that I T = mT . Let P ∈ Min(mR[It]) be such that P = Q ∩ R[It] and V = (R[It])Q ∩ Q(R) where Q ∈ Min(mR[It]). Then [

T R : ] = 1 mT m

⇐⇒

R[It] P

is regular.

M.-K. Kim / Journal of Algebra 488 (2017) 290–314

312

With Corollary 4.4 and Corollary 5.7, we observe the following. Remark 5.8. Let the notation be as in Setting 4.1. Let n ≥ 1 be an integer. Assume that I T = mnT . Consider the following statements: (1) n = 1. R (2) [ mTT : m ] = 1. (3)

R[It] P

is regular.

If any two items of (1), (2), and (3) are satisfied, then the remaining item holds.  Example  5.9 demonstrates that the quotient ring R1 R m1 : m > 1.

R[It] P

is not regular in the case where

Example 5.9. ([10, Example 6.11]) Let (R, m) be a 3-dimensional regular local ring with   R maximal ideal m = (x, y, z)R and k := m = Q. Let R1 := R m x N , where 1



 m y  z 2 . N1 := x, , −3 R x x x Let V be the order valuation ring of R1 with corresponding valuation v := ordR1 . Then we have  2 2 2 (1) v(x) = v( xy ) = v( xz − 3) = 1, and the images of xy2 , z −3x in the residue field x3 kv of V are algebraically independent over R/m. Also v(z 2 − 3x2 ) = 1 + v(x2 ) = 3. Therefore

v := ordR1

x

y

z

z 2 − 3x2

1

2

1

3

(2) I := PRR1 = (x3 , xy, z 2 − 3x2 , y 2 , yz, z 3 )R and we have PR0 R1

x3

xy

z 2 − 3x2

y2

yz

z3

v := ordR1

3

3

3

4

3

3

   2 (3) ordR (I) = 2 and IR1 = x2 I R1 , where I R1 = m1 = x, xy , xz − 3 R1 . (4) ReesR I = {ordR0 , ordR1 }. (5) Let P ∈ Min(mR[It]) be such that P = Q ∩ R[It] and V = W ∩ Q(R) with W := (R[It])Q where Q ∈ Min(mR[It]). Let t1 := x2 t. Then we have the following inclusion R[It] ⊂ R[It] ⊂ R1 [m1 t1 ] = R1 [m1 t1 ] ⊂ W

M.-K. Kim / Journal of Algebra 488 (2017) 290–314

313

Let P  be the center of W on R1 [m1 t1 ]. Then P  = m1 R1 [m1 t1 ], and hence G :=

R1 [m1 t1 ] W R[It] R1 [m1 t1 ] ⊂ =: B ⊂ = . P P m1 R1 [m1 t1 ] mW

Then we have:     I (a) edim R[It] = dim = 5 k IV . P     I1 1 [m1 t1 ] R1 (b) edim mR1 R = dim = 3 (I1 )V . 1 [m1 t1 ] m1     √ R1 I1 R1 R1 (c) dimk (II11)V = dimk ( m ) dim = Q( 3). = 2 · 3 = 6, where m (I ) 1 1 V 1 m1   R1 R (d) By Corollary 5.5, e(GN ) = e(BM ) m : = 1 · 2 = 2. Hence we have m 1 e(GN ) = 2 < 3 = 5 − 3 + 1 = edim(GN ) − dim(GN ) + 1. By [20, Exercise 11.10, p. 233], G = regular.

R[It] P

is not Cohen–Macaulay, and hence not

Acknowledgment The author is grateful to Professor William Heinzer for conversations about the topics discussed in this paper. References [1] S.S. Abhyankar, On the valuations centered in a local domain, Amer. J. Math. 78 (1956) 321–348. [2] W. Bruns, J. Herzog, Cohen–Macaulay Rings, revised edition, Cambridge Univ. Press, Cambridge, 1998. [3] S. Cutkosky, Factorization of complete ideals, J. Algebra 115 (1988) 144–149. [4] E. De Negri, Toric rings generated by special stable set of monomials, Math. Nachr. 203 (1999) 31–45. [5] J. Gately, ∗-Simple complete monomial ideals, Comm. Algebra 33 (2005) 2833–2849. [6] D. Grayson, M. Stillman, Macaulay2, a software system for research in algebraic geometry and computer algebra, http://www.math.uiuc.edu/Macaulay2/doc/Macaulay2-1.9.2/share/doc/Macaulay2/ Normaliz/html. [7] W. Heinzer, M.-K. Kim, Integrally closed ideals in regular local rings of dimension two, J. Pure Appl. Algebra 216 (2012) 1–11. [8] W. Heinzer, M.-K. Kim, Integrally closed ideals and Rees valuation, Comm. Algebra 40 (2012) 3397–3413. [9] W. Heinzer, M.-K. Kim, Complete ideals and multiplicities in two-dimensional regular local rings, J. Algebra 377 (2013) 250–268. [10] W. Heinzer, M.-K. Kim, The Rees valuations of complete ideals in a regular local ring, Comm. Algebra 43 (2015) 3249–3274. [11] W. Heinzer, M.-K. Kim, M. Toeniskoetter, Finitely supported *-simple complete ideals in a regular local ring, J. Algebra 401 (2014) 76–106. [12] M. Hochster, Rings of invariants of tori, Cohen–Macaulay rings generated by monomials, and polytopes, Ann. of Math. 96 (1972) 318–337. [13] C. Huneke, The primary components of and integral closures of ideals in 3-dimensional regular local rings, Math. Ann. 275 (1986) 617–635. [14] C. Huneke, J. Sally, Birational extensions in dimension two and integrally closed ideals, J. Algebra 115 (1988) 481–500.

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