THE GOTO NUMBERS OF PARAMETER IDEALS WILLIAMHEINZERANDIRENASWANSON 8 0 0 Abstract. LetQbeaparameteridealofaNoetherianlocalring(R,m). The 2 Goto number g(Q) of Q is the largest integer g such that Q :mg is integral over Q. Weexaminethevalues ofg(Q)asQvariesovertheparameter ideals n of R. We concentrate mainlyon the case where dimR=1, and many of our a resultsconcernparameteridealsofanumericalsemigroupring. J 4 ] C 1. Introduction A . This note started from the group work at the workshop “Integral closure, mul- h t tiplier ideals, andcores”that took place atthe AmericanInstitute of Mathematics a m (AIM) in PaloAlto, California,in December 2006. Shiro Goto presentedthe back- [ ground, motivation, and some intriguing open questions. 1 Recall that if (R,m) is a Noetherian local ring with dimR = d, then an m- v 5 primary ideal Q is called a parameter ideal if Q is generated by d elements. 8 6 A motivating result for the group work at AIM is: 0 . Theorem 1.1. (Corso, Huneke, Vasconcelos [2], Corso, Polini [4], Corso, Polini, 1 0 Vasconcelos [5], Goto [6]) Let (R,m) be a Cohen-Macaulay local ring of positive 8 0 dimension. Let Q be a parameter ideal in R and let I =Q:m. Then the following : v are equivalent: i X (1) I2 6=QI. r a (2) The integral closure of Q is Q. (3) R is a regular local ring and µ(m/Q)≤1. Consequently, if (R,m) is a Cohen-Macaulaylocalring that is not regular,then I2 =QI. If dimR>1, it follows that the Rees algebra R[It] is a Cohen-Macaulay ring, and even without the assumption that dimR > 1, the fact that I2 = QI implies that the associated graded ring gr (R) = R[It]/IR[It] and the fiber ring I R[It]/mR[It] are both Cohen-Macaulay. In [7], Goto, Matsuoka, and Takahashi explore the Cohen-Macaulayness and Buchsbaumness of the associated graded and fiber rings and Rees algebras for 2000 Mathematics Subject Classification. Primary13A15. 1 2 WILLIAMHEINZERANDIRENASWANSON ideals I = Q : m2 under the condition that I3 = QI2. They also give examples showingthatCohen-Macaulaynessdoesnotalwayshold. Notice thatthe condition I3 =QI2 implies that I is integralover Q, so I ⊆Q, where Q denotes the integral closure of Q [13, Corollary 1.1.8]. It seems that a natural next step would be to explore the Cohen-Macaulay property for the various ring constructs from the ideal I =Q:m3. We expect the necessity of even further restrictions on R and I. However, rather than examining each of I = Q : mi for increasing i in turn, we pass to examining I = Q : mg, where g is the greatest integer such that Q:mg is integral over Q. Because of the pioneering work Shiro Goto has done in this area we define the Goto number of a parameter ideal Q as follows: Definition 1.2. Let Q be a parameter ideal of the Noetherian local ring (R,m). Thelargestintegerg suchthatQ:mg isintegraloverQisdenotedg(Q)andcalled the Goto number of Q. In the case where dimR = 1 and Q = xR, we sometimes write g(x) instead of g(Q). Notice that the Goto number g(Q) is well defined, for Q : m0 = Q : R = Q is integral over Q, and for sufficiently large n, mn ⊆Q, so Q:mn =R, which is not integral over Q. During the workshop we concentrated on various invariants, dubbed “Goto in- variants of a Noetherian local ring (R,m)”, that involve the Goto numbers of pa- rameterideals. These invariantsare discussedinSection 2. During our subsequent work, we decided that the set G(R)={g(Q) | Q is a parameter ideal of R}, where R is a fixed one-dimensional Noetherian local ring is a possibly more inter- esting invariant. Most of the paper has to do with an examination of the integers that are in G(R). In the case where (R,m) is an arbitrary one-dimensional Noe- therian local ring, we prove the existence of a positive integer n such that every parameter ideal contained in mn has Goto number the minimal integer in G(R). With additional hypothesis on R, we prove that the set G(R) is finite. Our notation is mainly as in [13]. In particular,we use R to denote the integral closure of the ring R, and J to denote the integral closure of the ideal J of R. For many of the examples in the paper, the calculations were done using the symbolic computer algebra system Macaulay2 [8]. THE GOTO NUMBERS OF PARAMETER IDEALS 3 For muchofthe paperwe focus onaspecialtype ofone-dimensionalNoetherian local domain. As in the monograph of Ju¨rgen Herzog and Ernst Kunz [10], we consider a rank-one discrete valuation domain V with field of fractions K and let v : K \{0} → Z denote the normalized valuation associated to V. Thus if x ∈ V generates the maximal ideal of V, then v(x) = 1. Associated with each subring R of V is a subsemigroup G(R)={v(r) | r∈R\{0} } of the additive semigroup N 0 of nonnegative integers. G(R) is the value semigroup of R with respect to V. Definition 1.3. A subring R of V is called a numerical semigroup ring associated to V if it satisfies the following properties: (1) R has field of fractions K and the integral closure of R is V. (2) V is a finitely generated R-module. (3) There exists x ∈ V with v(x) = 1 such that xn ∈ R for each integer n such that n=v(r) for some r∈R, and if m=xV ∩R, then the canonical injection R/m֒→V/xV is an isomorphism. The value semigroup G(R) is the numerical semigroup associated to R Remark 1.4. Let R be a numerical semigroup ring associated to the valuation domain V as in (1.3). We then have the following. (1) SinceV isafinitelygeneratedR-module,Risaone-dimensionalNoetherian local domain with maximal ideal m [11, Theorem 3.7]. (2) Sincetheconductor[13,page234]ofRinV isnonzero,thevaluesemigroup G(R) = {v(r) | r ∈R\{0}} contains all sufficiently large integers. The largest integer f that is not in G(R) is called the Frobenius number of R, and C :=xf+1V is the conductor of R in V. (3) If 0 < a < a < ··· < a are elements of G(R) that generate G(R), then 1 2 d m=(xa1,xa2,...,xad)R. (4) An application of Nakayama’s lemma [11, Theorem 2.2] implies R[x]=V. (5) If u is a unit of V, then R/m= V/xV implies there exists a unit u of R 0 suchthatu−u ∈xV. Ifu6=u ,there existsa positiveintegeri suchthat 0 0 u−u = wxi, where w is a unit of V. Repeating the above process on w, 0 we see that every unit u of V has the form u=u +u x+···+u xf +α, 0 1 f where α∈C, u is a unit of R, and each u , 1≤i≤f, is either zero or a 0 i unit of R. 4 WILLIAMHEINZERANDIRENASWANSON (6) Every nonzero element r ∈ R has the form r = uxb for some b ∈ G and some unit u∈V. Multiplying u by a unit in R and using item (5), we see that every nonzero principal ideal of R has the form uxbR, where u=1+u x+u x2+···+u xf +α, 1 2 f where α∈C and each u is either zero or a unit of R. Thus i uxb =(1+u x+u x2+···+u xf)xb+αxb. 1 2 f Since uxb ∈ R, it follows that b+i ∈ G for each i such that u 6= 0. Also i α∈C implies α=uβ, where β ∈C. Thus uxb−αxb =uxb−uβxb =uxb(1−β). Since 1−β is a unit of R, we conclude that eachnonzero principal ideal of R has the form (1+u x+···+u xf)xbR, where b ∈ G, each u is either 1 f i zero or a unit of R, and if u 6=0, then b+i∈G. i (7) With r =uxb, if we pass to integral closure, we have (r)=(r)V ∩R=(xb)V ∩R=(xe :e∈G,e≥b)R. Remark 1.5. With additional assumptions about the rank-one discrete valuation domain V it is possible to realize numerical semigroup rings by starting with the group. Let k be a field and let x be an indeterminate over k. If V is either the formalpower seriesring k[[x]] orthe localizationof the polynomialringk[x] atthe maximal ideal generated by x, then for each subsemigroup G of N that contains 0 all sufficiently large positive integers, there exists a numerical semigroup ring R associated to V such that G(R)= G. In each case one takes generators a ,...,a 1 d for G. If V is the formal power series ring k[[x]], then R = k[[xa1,...,xad]] is the subring of k[[x]] generated by all power series in xa1,...,xad, while if V is k[x] localized at the maximal ideal generated by x, then R is k[xa1,...,xan] localized at the maximal ideal generated by xa1,...,xad. We observe in Proposition 1.6 a useful result for computing Goto numbers of parameter ideals in dimension one. Proposition 1.6. Let Q and Q be ideals of a Noetherian local ring (R,m). 1 2 Assume that Q is not contained in any minimal prime of R. If e is a positive 2 integer such that Q : me is not integral over Q , then Q Q : me is not integral 1 1 1 2 over Q Q . 1 2 THE GOTO NUMBERS OF PARAMETER IDEALS 5 Proof. It suffices to check integral closure modulo each minimal prime ideal, so we may assume that R is an integral domain [13, Proposition1.1.5]. Let x∈Q :me. 1 Then xQ ⊆ Q Q : me. If all the elements in xQ are integral over Q Q , then 2 1 2 2 1 2 [13, Corollary 6.8.7] implies that x is integral over Q . (cid:3) 1 Indimensionone,theproductoftwoparameteridealsisagainaparameterideal. Thus Proposition 1.6 has the following immediate corollary. Corollary 1.7. Let (R,m) be a one-dimensional Noetherian local ring. If Q and 1 Q are parameter ideals of R, then g(Q Q )≤min{g(Q ),g(Q )}. 2 1 2 1 2 A strict inequality may hold in Corollary 1.7 as we illustrate in Example 1.8. Example 1.8. Let G = h3,5i be the numerical subsemigroup of N generated 0 by 3 and 5, and let R as in Remark 1.5 be a numerical semigroup ring such that G(R)=h3,5i. A direct computation shows that the parameter ideal Q=x5R has Goto number g(x5) = 3, while Q2 = x10R has the property that x9 ∈ x10R : m3. Therefore x10R:m3 is not integral over x10R and g(x10)=2. The Goto numbers of parameter ideals of a Gorenstein local ring may be de- scribed using duality as in Proposition 1.9. Proposition 1.9. Let Q be a parameter ideal of a Gorenstein local ring (R,m). Assume that Q(Q. Let J =Q:Q. Then g(Q)=max{i | J ⊆mi+ Q}. Proof. Since R/Q is a zero-dimensional Gorenstein local ring, (Q : J) = Q, and (Q:mi)⊆Q if and only if J ⊆mi+ Q, cf. [1, (3.2.12)]. (cid:3) 2. Goto invariants of local rings need not be bounded Since a regular local ring of dimension one is a rank-one discrete valuation do- main, the Goto number of every parameter ideal is 0 in this case. We prove below thatin a two-dimensionalregularlocalring,the Goto number of a parameterideal Q is precisely ordQ−1, where ordQ is the highest power of m that contains Q. Thusinatwo-dimensionalregularlocalring,theGotonumberofaparameterideal is uniquely determined by the order of the parameter ideal. It seems natural to expect at least for many local rings (R,m) that the Goto number g(Q) becomes 6 WILLIAMHEINZERANDIRENASWANSON largerasQisinhigherandhigherpowersofm. Thefollowingareseveralinvariants of a local ring (R,m) involving Goto numbers g(Q) of parameter ideals Q of R. g(Q) goto (R) = sup | Q varies over parameter ideals of R , 1 (cid:26)ord(Q) (cid:27) g(Q) goto (R) = sup | Q varies over parameter ideals of R , 2 (cid:26)ord(Q:m) (cid:27) g(Q) goto (R) = sup | Q varies over parameter ideals of R . 3 (cid:26)ord(Q:mg(Q)) (cid:27) In order to avoid division by zero, in the definition of goto (R), we exclude the 2 case where R is a regular local ring and Q=m. Example2.1demonstratestheexistence,foreveryintegerd≥3,ofaregularlocal ring (R,m) of dimension d for which each of the invariants goto (R),i ∈ {1,2,3}, i is infinity. Example 2.1. Let k be a field, d an integer > 2, x ,...,x variables over k. Let 1 d n≥e be positive integers, and let Q=(xe,xn,...,xn). Then g(Q)= (d−2)(n− 1 2 d 1)+e−1. For we have: (xe,xn,...,xn):(x ,...,x )(d−2)(n−1)+e−1 =(xe)+(x ,...,x )n, 1 2 d 1 d 1 1 d which is integral over Q, and (xe,xn,...,xn):(x ,...,x )(d−2)(n−1)+e 1 2 d 1 d contains xn−1, which is not integral over Q. Furthermore, 2 ord(Q)=ord(Q:m)=ord(Q:m(d−2)(n−1)+e−1)=e. Thus, for each i ∈ {1,2,3}, goto (R) ≥ (d−2)(n−1)+e−1 for all n ≥ e. Since d > 2, i e we have goto (R)=∞. i In the case where (R,m) is a two-dimensional regular local ring, we prove in Theorem 2.2 that the Goto number of a parameter ideal Q depends only on the order of Q. Theorem 2.2. Let (R,m) be a two-dimensional regular local ring. Then for each parameter ideal Q of R, the Goto number g(Q)=ord(Q)−1. Proof. Passing to the faithfully flat extension R[X]mR[X] preserves the parameter ideal property and its order and Goto number, so that without loss of generality we may assume that R has an infinite residue field. Let k = ordQ. The proof of THE GOTO NUMBERS OF PARAMETER IDEALS 7 [14, Theorem 3.2] shows that k−1 ≤ g(Q). (In Wang’s notation in that proof, it is shown that Q : mk−1 ⊆ (Qmk−1 : mk−1) ⊆ Q.) Now we prove that g(Q) ≤ k − 1. Let Q = (a,b). Let x ∈ m\m2 be such that ord(a) = ord(a(R/(x))) and ord(b) = ord(b(R/(x)). Since the residue field of R is infinite, it is possible to find such an element x. The condition needed for x is that its image in the associatedgraded ring grm(R) is not a factor of the images of a and b in grm(R)). SinceR/(x)isaone-dimensionalregularlocalring,henceaprincipalidealdomain, by possibly permuting a and b we may assume that b ∈ (a,x). By subtracting a multiple of a from b, without loss of generality b=b x for some b ∈R. Note that 0 0 (a,x)=mk+(x), andorda=ordQ=k ≤ordb. Itfollowsthatb mk ⊆b (a,x)⊆ 0 0 (a,xb ) ⊆ (a,b). However, b 6∈ (a,b): otherwise for all discrete valuations v 0 0 centered on m, v(b ) ≥ min{v(a),v(b)}, whence since v(b) = v(b x) > v(b ), 0 0 0 necessarily v(b ) ≥ v(a) for all such v, so that b ∈ (a) = (a), contradicting the 0 0 assumptionthat(a,b)isaparameterideal. Thisprovesthatg(Q)<k=ordQ. (cid:3) If (R,m) is a regular local ring, then the powers of m are integrally closed. Hence, in this case, if Q : mi is integral over Q, then ordQ = ord(Q : mi). Thus Theorem 2.2 implies the following: Corollary 2.3. If (R,m) is a two-dimensional regular local ring, then each of the invariants goto (R),i∈{1,2,3}, is one. i Remark 2.4. Let (R,m) denote the m-adic completion of the Noetherian lo- cal ring (R,m). Sincbe bR/I ∼= R/IR for each m-primary ideal I of R, the m- primary ideals of R are in one-to-bonebinclusion preservingcorrespondencewith the m-primary ideals of R. Also, if I is an m-primary ideal, then IR is the integral cblosure of IR [13, Lebmma 9.1.1]. Since R/m ∼= R/m, and sinbce each param- eter ideal ofbR has the form QR, where Q is a pbarabmeter ideal of R, the set {ℓR(Q/Q) | Qbis a parameter ideabl of R} is identical to the corresponding set for R. Since R is flat over R, we also have (Q :R mi)R = (QR :Rb mi) for each pbositive intbeger i. Therefore, for each parameter idealbQ of R,bthe Gcoto number g(Q) = g(QR). Hence the set G(R) = {g(Q) | Q is a parameter ideal of R} is identical to thbe corresponding set G(R) for R. b b 8 WILLIAMHEINZERANDIRENASWANSON 3. One-dimensional Noetherian local rings Throughout this section, let (R,m) be a one-dimensionalNoetherian local ring. In subsequent sections we restrict to the special case where R is a numerical semi- groupring. IfRisaregularlocalring,thenitisaprincipalidealdomain,andhence the Goto number g(Q) = 0 for every parameter ideal Q. Thus to get more inter- estingvariationsonthe Gotonumber ofparameterideals,werestrictourattention to non-regular one-dimensional Noetherian local rings. Corollary 1.7 is useful for examining the Goto number of parameter ideals. We observe in Theorem 3.1 that the Goto number of parameter ideals in a sufficiently high power of the maximal ideal of R are all the same and that this eventually constant value is the minimal possible Goto number of a parameter ideal of R. Theorem 3.1. Let (R,m) be a one-dimensional Noetherian local ring. (1) If yR is a parameter ideal of R, then g(Q)≤g(y) for every parameter ideal Q such that Q⊆yR. (2) There exists a positive integer n such that all parameter ideals of R con- tained in mn have the same Goto number. Moreover, this number is the minimal Goto number of a parameter ideal of R. Proof. If Q = qR is a parameter ideal and Q ⊆ yR, then q = yz for some z ∈ R. If Q = yR, then g(Q) = g(y), while if Q is properly contained in yR, then zR is a parameter ideal, and Corollary 1.7 implies that g(Q) ≤ g(y). This establishes item (1). For item (2), let yR be a parameter ideal such that g(y) is the minimal element of the set G(R)={g(Q) | Q is a parameter ideal of R}. Since yR is aparameterideal,there existsa positiveintegern suchthat mn ⊆yR. By item (1), g(Q) ≤ g(y) for every parameter ideal Q ⊆ mn, and by the choice of g(y), we have g(Q) = g(y) is the minimal Goto number of a parameter ideal of R. (cid:3) Remark 3.2. Let g = g(Q) denote the Goto number of the parameter ideal Q. The chain of ideals Q=Q:m0 (Q:m(Q:m2 (···(Q:mg ⊆Q THE GOTO NUMBERS OF PARAMETER IDEALS 9 implies that the length ℓ (Q/Q) of the R-module Q/Q is an upper bound on R g(Q). Thus if (R,m) is a one-dimensional Noetherian reduced ring1 such that R is a finitely generated R-module, then the length of R/R is an upper bound for g(Q) and therefore the set G(R) is finite. To see this, let Q = qR be a parameter ideal of R. Then qR is an integrally closed ideal of R, and Q = qR∩R, cf. [13, Proposition 1.6.1]. Thus we have ℓ (R/R)=ℓ (qR/qR)≥ℓ (Q/Q). R R R Remark 3.3. For certain parameter ideals Q it is possible to compute the Goto number g(Q) as an index of nilpotency. If Q =xR is a reduction of m, then m is the integral closure of Q and g(Q)= max{i | (Q:mi)6=R}= min{i | mi+1 ⊆Q} is the index of nilpotency of m with respect to Q [9, (4.4)]. This is an integer that islessthanorequalto the reductionnumberofm withrespecttoQ,withequality holding if the associated graded ring grm(R) is Cohen-Macaulay. We prove in Theorem 3.4 a sharpening of Theorem 3.1 in the case where R is module-finite over R. Theorem 3.4. Let(R,m)beaone-dimensional Noetherian localreducedringsuch that R is module-finite over R. Let C =R: R be the conductor of R in R, and let R x∈m and y ∈C generate parameter ideals. Then for each positive integer n, the Goto number g(xny) = g(xy). Thus for all parameter ideals Q = qR ⊆ xC = xC, we have g(Q) = g(xy). Furthermore, this is the minimal possible Goto number of a parameter ideal in R. Proof. By Corollary 1.7, g(xy)≥g(xny). To prove that g(xy)≤g(xny), it suffices to prove for each positive integer i that (1) (xyR:mi)⊆xyR =⇒ (xnyR:mi)⊆xnyR. Assume there exists w ∈ R with wmi ⊆ xnyR and with w ∈/ xnyR. Notice that xw ∈ xnyR ⊆ xnC ⊆ xnR implies w ∈ xn−1R. Therefore by replacing w if necessary by wxj for some positive integer j, we may assume that w ∈ xn−1R, so 1If (R,m) is a Noetherian local ring that is not equal to its total quotient ring and if R is module-finiteoverR,thenRisreduced. Forifx∈misaregularelementandy∈Risnilpotent, theny/xn∈R,soy∈xnR,foreachn∈N. ButifRismodule-finiteoverR,thenRisNoetherian andT∞n=1xnR=(0),cf. [13,Prop. 1.5.2]. 10 WILLIAMHEINZERANDIRENASWANSON w = xn−1z for some z ∈ R. Thus xn−1zmi ⊆ xnyR implies that zmi ⊆ xyR, so z ∈xyR:mi. Moreover,w =xn−1z ∈/ xnyRimpliesthatz ∈/ xyR. Thisestablishes the implication displayed in (1). Theorem 3.1 implies that for n sufficiently large g(xny) is the minimal Goto number of a parameter ideal of R. This completes the proof of Theorem 3.4. (cid:3) In comparisonwith Theorem3.4, we demonstratein Example4.6 thatthe Goto number g(Q)of parameterideals containedinthe conductor need notbe constant, even in the case where (R,m) is a Gorenstein numerical semigroup ring. Theorem 3.5 establishes conditions on a one-dimensional Noetherian local ring R in order that the set {ℓ (Q/Q) | Q is a parameter ideal of R} is finite. R Theorem 3.5. Let (R,m) be a one-dimensional Noetherian local ring, let (R,m) denote the m-adic completion of R, and let n denote the nilradical of R.bTbhe following statements are equivalent. b (1) The length ℓb(n) is finite. R (2) The set {ℓ (Q/Q) | Q is a parameter ideal of R} is finite. R Proof. ByRemark2.4, item(2)holdsforRifandonlyifitholdsforR. Therefore, to prove (1) ⇐⇒ (2), we may assume that R is complete. b Assume that ℓ (n) is finite, and let R′ = R/n. If Q is a parameter ideal R of R, then n ⊂ Q and ℓ ((Q + n)/Q) ≤ ℓ (n). Since R′ is a reduced com- R R plete Noetherian local ring, its integral closure is a finite R′-module. Thus by Remark 3.2, the set {ℓ′ (QR′/QR′) | Q is a parameter ideal of R′} is bounded by R someintegers. Itfollowsthats+ℓ (n)isanupper boundforℓ (Q/Q),sothe set R R {ℓ (Q/Q) | Q is a parameter ideal of R} is finite. R Assume that ℓ (n) is infinite and let Q = xR be a parameter ideal of R. For R 1 each positive integer n, let Q =xnR. Then Q +n⊆Q , and n n n (Q +n) n n n ∼= = . Q (Q ∩n) Q n n n n Hence ℓ (Q /Q )≥ℓ (n/xnn). Therefore ℓ (Q /Q ) goes to infinity as n goes R n n R R n n to infinity. This completes the proof of Theorem 3.5. (cid:3) Corollary 3.6. Withnotation as in Theorem 3.5, if thelengthℓb(n)is finite, then R the set G(R)={g(Q) | Q is a parameter ideal of R} is finite. Proof. Apply Theorem 3.5 and Remark 3.2. (cid:3)