Multiplicity and Tight Closures of Parameters 1

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1 Journal of Algebra 244, (2001) oi: /jabr , available online at on Multiplicity an Tight Closures of Parameters 1 Shiro Goto an Yukio Nakamura Department of Mathematics, School of Science an Technology, Meiji University, , Japan goto@math.meiji.ac.jp, ynakamu@math.meiji.ac.jp Communicate by Eva Bayer Fluckiger Receive January 31, 2001 In this paper we investigate the relation between the multiplicity an the tight closure of a parameter ieals. We shall show that local rings having a parameter ieal whose multiplicity agrees with the colength of its tight closure are Cohen Macaulay rings Acaemic Press Key Wors: Cohen Macaulay ring; F-rational ring; multiplicity; tight closure. 1. INTRODUCTION Let R be a Noetherian local ring with the maximal ieal an assume that R possesses positive characteristic p>0. For each -primary ieal I in R lete I R an I enote the multiplicity of R with respect to I an the tight closure of I, respectively. Here let us briefly recall the efinition of tight closures. For an ieal in R let q = a q a R where q = p e with e 0. Let R 0 = R\ min R. Let enote the set of elements x R for which there exists c R 0 such that cx q q for all q 0. Then the set forms an ieal in R containing, which we call the tight closure of. The ieal is sai to be tightly close if =. With this notation one always has the inequalities l R R/ e q R an l R R/ l R R/ for any parameter ieal in R. In this paper we are intereste in the problem of which is bigger, e R or l R R/. About this problem Watanabe an Yoshia [WY] pose the following conjecture. Let R enote the -aic completion of R. 1 Both authors are supporte by a Grant-in-Ai for Scientific Research, Japan /01 $35.00 Copyright 2001 by Acaemic Press All rights of reprouction in any form reserve. 302

2 Conjecture 1 [WY, Conjecture 1.6]. Ass R = Assh R. Then multiplicity of parameters 303 Suppose R is unmixe; that is, (1) e R l R R/ for every parameter ieal in R. (2) If e R =l R R/ for some parameter ieal in R, then R is a Cohen Macaulay F-rational local ring. Here we putassh R = Spec R im R = im R/. We say that R is F-rational if every parameter ieal is tightly close. Assertion (2) in Conjecture 1 also tells us that, once e R =l R R/ for some parameter ieal in R, then e R =l R R/ hols true for every parameter ieal in R. In aition, accoring of Feer an Watanabe s result (cf. [FW, Proposition 2.2]). The F-rationality of R follows from the existence of at least one parameter ieal being tightly close when R is Cohen Macaulay. Therefore, in assertion (2) in Conjecture 1, it is essential to prove the Cohen Macaulayness of R from the existence of a parameter ieal satisfying e R =l R R/. In [WY] Watanabe an Yoshia stuie Hilbert Kunz multiplicities e HK R for local rings R an prove that every unmixe local ring R must be regular if e HK R =1. In their proof they gave a nice estimation of e R an l R R/ an showe that assertions (1) an (2) in Conjecture 1 hol true for a special kin of parameter ieals in unmixe local rings. Because the local ring R is so-calle FLC if an only if R contains at leastone parameter ieal of this kin, from their result it irectly follows that assertions (1) an (2) in Conjecture 1 hol true, for instance, if R is a Buchsbaum local ring of positive epth. However, when R is not unmixe, assertions (1) an (2) in Conjecture 1 o not necessarily hol true (Example 2.3). The purpose of this paper is to announce that Conjecture 1 hols true for a huge class of rings. Theorem 1.1. Let R be a homomorphic image of a Cohen Macaulay local ring of characteristic p > 0 an assume that R is equiimensional. Then assertion (1) in Conjecture 1 hols true. We say that R is equiimensional if im R = im R/ for all min R. To get assertion (2), we nee a stronger assumption for R, butitis much weaker than the assumption of R being FLC or Buchsbaum. Theorem 1.2. Let R be a homomorphic image of a Cohen Macaulay local ring of characteristic p>0 an assume that Ass R = Assh R. Then assertion (2) in Conjecture 1 hols true. Our theorems not only give affirmative answers for the original problem but also recognize the utility of the tight closure technique in the analysis of rings of positive characteristic as well as it simplifies the proof of the

3 304 goto an nakamura main result in [WY]. Also, both theorems suggest that the behavior of e R l R R/ gives much information for local rings R. The analysis of the ifference e R l R R/ is very closely relate to that of l R /. In the coming paper [GN], we shall stuy an supremum of l R / when the ieal runs over all parameter ieals in R an fin the conition uner which the supremum is finite. Next, we state how this paper is organize. Section 2 is evote to giving the proof of Theorem 1.1. Among the proof, the concepts of filter regular sequence an colon capturing property of tight closure play a key role. We also in Section 2 give an example of a non-unmixe local ring R an parameter ieals in R satisfying l R R/ > e R an l R R/ = e R. Section 3 is evote to giving the proof of Theorem 1.2. To show Theorem 1.2, we nee to make use of the sequence satisfying a stronger property than just filter regular sequences. We shall state how to choose such a sequence an give several conclusions such sequences yiel. One of the ifficulties of the argument of tight closure comes from the fact that tight closure is not preserve in the factor ring in general. However, we shall fin, in the proof of Theorem 1.1, that the tight closure of the ieal generate by a subsystem of parameters is very close to the unmixe componentof thatieal. Therefore, we can occasionally reuce the augmentof tight closure to the case of imension two. It is a key of the proof of the Theorem 1.2. Throughoutthis paper let R enote a Neotherian local ring of imension. For a finitely generate R-moule M we enote by l R M the length of M an by e I M the multiplicity of M with respect to an -primary ieal I. LetH i M i stan for the ith local cohomology moule of M with respect to. 2. PROOF OF THEOREM 1.1 Let I be an ieal in the local ring R. We putu I = n>0 I n an call itthe unmixe componentof I. We recall the efinition of a filter regular sequence. A sequence a 1 a 2 a s of elements of R is calle a filter regular sequence if a 1 a 2 a i R a i+1 U a 1 a 2 a i R for all 0 i<s.let be a parameter ieal in R. One can always choose a system a 1 a 2 a of generators of which forms a filter regular sequence. In fact, let i 0 an suppose that a 1 a 2 a i is partof minimal generators of an forms a filter regular sequence. Take a i+1 such that a i+1 oes notbelong to + a 1 a 2 a i R nor any Q in Ass R R/ a 1 a 2 a i R\. Then itreaily follows thatthe sequence a 1 a 2 a i is a filter regular sequence. We put i = a 1 a 2 a i R for a parameter ieal generate by a 1 a 2 a.

4 multiplicity of parameters 305 The following property is calle the colon capturing property of tight closure ue to Hochster an Huneke (cf. [HH, Theorem 7.9; BH, Theorem ] for the proof). It plays a key role throughout this paper. Theorem 2.1 (Colon Capturing Property of Tight Closure). Let R be a homomorphic image of a Cohen Macaulay local ring of characteristic p>0 an assume that R is equiimensional. Let a 1 a 2 a be a system of parameters of R. Then, a 1 a 2 a i R a i+1 a 1 a 2 a i R for all 0 i< Proof of Theorem 1.1. Let be a parameter ieal of R. Choose a system a 1 a 2 a of generators of which is a filter regular sequence. Then we have e R =e a R/ 1. Besies, itfollows that e R =e a R/U 1 = l R R/U 1 +a R because U 1 / 1 is length finite an R/U 1 is a Cohen Macaulay ring of imension 1. Furthermore, we have U 1 = 1 a r for a suitable integer r. Thus, we getu 1 1, by the colon capturing property. Eventually, we get U 1 +a R, whence the inequality of Theorem 1.1 follows. Remark 2.2. As can be seen in the proof of Theorem 1.1, when the generator a 1 a 2 a of a parameter ieal is a filter regular sequence, the equality e R =l R R/ hols true if an only if = U 1 +a R. To illustrate our results let us note the following. Example 2.3. Let A = k X Y Z be the formal power series ring of three variables X Y Z over a fiel k of characteristic p>0. Let R = A/ XY XZ =k x y z. Then epth R = 1 butthe local ring R is not unmixe. Let us take a parameter ieal = a b in R so that neither a nor b is a zeroivisor in R. Then the following two assertions hol true. (1) l R R/ e R, (2) l R R/ =e R if an only if + y z =. For instance, the equality l R R/ =e R hols true if = x + y x + z, but l R R/ > e R if = x 2 + y x 2 + z. Proof. Let R be the normalization of R in its quotient ring. We may write R = R 1 R 2, where R 1 = R/xR = k Y Z an R 2 = R/ y z R = k X. Notice that R = R. In fact, R R since R 0 R 0, while R = R since R is regular ring (hence every ieal is tightly

5 306 goto an nakamura close). Besies, we have ar = ar (the integral closure of ar) by [HH, Corollary 5.8] an ar = a R since a R R. Hence, we have R = R = a R + b R = ar + br Consequently, it follows that = R. On the other han, l R R/ =l R R/ R = l R R/ R 1 since l R R/R =1 = l R1 R 1 / R 1 +l R2 R 2 / R 2 1 = e R 1 +l R2 R 2 / R 2 1 Because e R =e R 1, we always get the inequality l R R/ e R. The equality hols true if an only if l R2 R 2 / R 2 =1. This conition is equivalentto R 2 = R 2 an also equivalentto = + y z R. Therefore, the example follows. 3. PROOF OF THEOREM 1.2 We begin with the following. Lemma 3.1. Suppose that R is a complete local ring. Let K R be the canonical moule of R an S = Hom R K R K R. Let ϕ R S be the canonical map; i.e., ϕ a x =ax for a R an x K R.IfAss R Assh R, then we have the following statements. (1) ker ϕ is finite length. (2) H 1 R is isomorphic to H0 (Coker ϕ). In particular, H1 R is finite length. Proof. (1) Let U = Ker ϕ. Suppose that Ass R U\. Then we have K R = K R an S = Hom R K R K R = R since Assh R. It implies that U = 0, an that contraicts the choice of. Hence, U is finite length. (2) Let C = Coker ϕ. Divie the exact sequence 0 U R S C 0 into two short exact sequences: 0 U R ϕ R 0 an 0 ϕ R S C 0. From the firstsequence itfollows that H 1 R = H 1 ϕ R since U is finite length, while from the secon one we geth 0 C = H 1 ϕ R since epth R S 2. Hence, H1 R = H 0 C. Lemma 3.2. Let R be a homomorphic image of a Cohen Macaulay local ring an assume that Ass R Assh R. Then is a finite set. F = Spec R ht R > 1 = epth R

6 multiplicity of parameters 307 Proof. Let F = P Spec R ht RP >1 = epth R P P R. Then F P R P F. In fact, let F an take P min R/ R. One can reaily check that ht R P>1, epth R P = 1, an = P R. Hence P F an itfollows thatf P R P F. So it is enough to show that F is a finite set. Here, we note that Ass R Assh R R. In fact, take P Ass R with P R, an there exists Ass R with such that P/ R Ass R/ R. From our assumption, it follows that Assh R, whence im R/ = im R/ R =. Furthermore, we have Ass R/ R = Assh R/ R because R is a homomorphic image of a Cohen Macaulay ring (hence R is unmixe). Thus, im R/P = an P Assh R. By the above argument, we may assume that R is a complete local ring. Let R S = Hom R K R K R be the canonical map as in Lemma 3.1. Then, ityiels the exactsequence 0 U R S C 0. Localize it at F. Then, we getthe shortexactsequence 0 R S C 0 since U is finite length. By the choice of, we have epth R = 1 an epth R S 2. Thus, epth R C = 0, whence Ass R C. This implies that F Ass R C an F is finite. To prove Theorem 1.2, we nee to choose a better generating system for parameter ieals. The following proposition inicates the choice. Proposition 3.3. Let R be a homomorphic image of a Cohen Macaulay local ring an assume that Ass R Assh R. Let be a parameter ieal. Then,there exists a system a 1 a 2 a of generators of such that Ass R/ i Assh R/ i for all 0 i. Proof. If = 0, we have nothing to prove. Let >0. Repeating the same proceure, it is enough to prove the statement in the case where i = 1. Let F be as in Lemma 3.2. Choose a 1 so that ( ) ( ) a 1 \ Assh R Take Ass R R/a 1 R with. Then epth R /a 1 R = 0, while epthr > 0 since Ass R. Hence epth R = 1. Itimplies thatht R = 1 since F. Thus, we have im R/ = im R ht R = 1 = im R/a 1 R. (Notice that R is a catenary an equiimensional ring.) Hence Assh R/a 1 R. This completes the proof. Corollary 3.4. Let R an a 1 a 2 a be as in Proposition 3.3. Then (1) a 1 a 2 a is a filter regular sequence. (2) a 1 a a 1 is also a filter regular sequence. (Here we exchange a 1 an a.) F

7 308 goto an nakamura Proof. (1) It is enough to check that 0 a 1 U 0 because we can repeat the same proceure for the factor ring R/a 1 R. Besies, itis enough to prove that 0 a 1 if finite length because U 0 =H 0 R. Take Ass R 0 a 1 with. Then a 1 as Assh R. Hence 0 a 1 = 0 an this is a contraiction. It implies that 0 a 1 is finite length. (2) The conition for a 1 a in Proposition 3.3 says nothing when i = 1. So the sequence a 1 a a 1 satisfies the same property. A Noetherian local ring R is calle an FLC ring (or a generalize Cohen Macaulay ring) if H i R has finite length unless i = im R. Itis known that if R is an FLC ring, then every system of parameters of R forms a filter regular sequence. The converse is also true if R is a homomorphic image of a Cohen Macaulay local ring (cf. [CTS]). Corollary 3.5. Let R be a homomorphic image of a Cohen Macaulay local ring an assume that Ass R Assh R. If = 2,then R is FLC. Proof. Every system of parameters satisfies the conition of Proposition 3.3 when = 2. So itforms a filter regular sequence by Corollary 3.4. Hence R is FLC. The next is also an important corollary to Proposition 3.3. Corollary 3.6. Let R be a homomorphic image of a Cohen Macaulay local ring an assume that Ass R Assh R. Let a 1 a 2 a be a generating system of as in Proposition 3.3. Let 2. IfH 1 R/ 2 = 0, then H i R = 0 for i = Proof. We may assume that 3. Because the assumption of the corollary is inherite to the factor ring R/ i, it is enough to prove the following statement; Let a R be a filter regular sequence of length one. Let 3. If H i R/aR = 0 for i = 1 2 2, then Hi R = 0 for i = Recalling that Ass R Assh R R (cf. Proof of Lemma 3.2), we may assume that R is a complete local ring. Hence H 1 R is finite length by Lemma 3.1. Because 0 a is finite length, the exact sequence 0 0 a R a R R/aR 0 yiels the following exact sequences: 0 0 a H 0 R H a 0 R H0 R/aR H1 R H a 1 R 0 0 H i R H a i R for i = 2 3 1

8 multiplicity of parameters 309 Hence, itfollows thath 1 R = 0 from the first sequence by Nakayama s lemma. On the other han, it follows that H i R = 0 for i = 2 3, 1 from the secon sequence because a power of a kills any elementof H i R. Thus, the corollary follows. Accoring to Corollary 3.6, in orer to see that R is Cohen Macaulay, it is enough to show that H 1 R/ 2 = 0. Tightclosures are notnecessarily preserve in the factor ring R/ 2. However, some property of the ring R that enables us to prove Theorem 1.2 is erive from the equality n>0 U 1 +a R = an preserve to the factor ring R/ 2.It reuces the problem to the case where im R = 2 an the key in the case of imension two is the following. Lemma 3.7. Let R be an FLC local ring of imension 2 an let a b be a system of parameters of R. Assume that the equality U ar +b n R = ar + U b n R hols true for all n 0. Then H 1 R = 0. Proof. Passing to R/U 0, we may assume that epth R>0. In fact, one can reaily check that R/U 0 is FLC, U ar /U 0 =U ar + U 0 /U 0, an H 1 R =H1 R/U 0. Then by Theorem 5.6 of [GY], there exists a Cohen Macaulay R-moule E such that R E an im R E = 2. Notice that U ar ae because ar R n ae E n an because a b is a regular sequence on E. Also, notice that E/R = H 1 R. Take an integer n so that b n E/R = 0. Then U ar =U ar U ar +b n R = U ar ar + U b n R = ar + U ar U b n R ar + ae b n E = ar + ab n E ar Hence we obtain that U ar =ar an this implies epth R/aR > 0. Thus R is Cohen Macaulay since a is a non-zero ivisor on R. The lemma follows. We are now in a position to finish the proof of Theorem 1.2. Let us recall the statement. Theorem 1.2. Let R be a homomorphic image of a Cohen Macaulay local ring of characteristic p>0 an assume that Ass R = Assh R. Then R is a Cohen Macaulay F-rational ring if the equality e R =l R R/ hols true for some parameter ieal.

9 310 goto an nakamura Proof. It is enough to show that R is Cohen Macaulay, since the F-rationality of R follows from the Cohen Macaulayness of R (cf. [FW]). We may assume 2. Let be a parameter ieal an suppose that e R =l R R/. We take a system a 1 a 2 a of generators of as in Proposition 3.3. Then = U 1 +a R by Remark 2.2 an Corollary 3.4. We furthermore have the following. Claim 1. Let n 1 be an integer an put J = a 1 a 1 a n. Then J = U 1 +a n R. Proof of Claim 1. The inclusion J U 1 +a n R follows from the colon capturing property. (See the proof of Theorem 1.1.) Let us prove the opposite inclusion. The proof will be one by inuction on n. Ifn = 1, there is nothing to prove. Assume that the claim hols true for n 1. Let x J. By the hypothesis of inuction, J U 1 +a n 1 R. So one can write x = y + a n 1 z, where y U 1 an z R. Notice that y 1 by the colon capturing property. Choose c R 0 so that cx q J an also cy q q 1 for all q 0. Then there exists w R such that cxq a nq w mo aq 1 aq 2 aq 1 R, because J q = a q 1 aq 1 anq R. Onthe other han, cx q ca n 1 q z q mo a q 1 aq 2 aq 1 R, because cxq = cy q + ca n 1 q z q an cy q q 1. Therefore, we get a n 1 q cz q a q w aq 1 aq 2 aq 1 R a By the assumption that R is a homomorphic image of a Cohen Macaulay ring, we may put R = A/I, where A is a Cohen Macaulay local ring of characteristic p an I is an ieal of A with ht A I = 0. Let x A be an inverse image of x R. Take ỹ z, an so on, similarly. Furthermore, choosing suitable elements, we may assume that ã 1 ã forms a regular, sequence on A. In fact, we can take them as a system of parameters of A (cf., e.g., [BH, Lemma ]). Taking the inverse image of the whole of (a) in A, we have ã n 1 q c z q ã q w ãq 1 ãq 2 ãq 1 A + I On the other han, we take an element A\ min A A/I an a power q = p e such that I q = 0. The choice is possible because R is an equiimensional an ht A I = 0 (cf., e.g., [BH, proof of Theorem ]). Then we have ã n 1 q c z q ã q w q ã qq 1 ãqq 2 ãqq 1 A Since ã 1 ã is a regular sequence on A, we can take off ã n 1 q from the above, whence we get c q z qq ã qq w q ã qq 1 ãqq 2 ãqq 1 A

10 Therefore, itfollows that multiplicity of parameters 311 c q z qq ã qq 1 ãqq 2 ãqq A for all q 0 b Taking the image of the whole of (b) in R, we have that z because the image of c q in R belongs to R 0. Now, we have x = y + a n 1 z U 1 +a n 1, an recall that = U 1 +a R. Combining them, we get x U 1 +a n R. Thus, our claim follows. We return the proof of the theorem. By Remark 2.2 an Claim 1, we have the equality e J R =l R R/J for J = a 1 a 1 a n R. Onthe other han, the sequence a 1 a 2 a n a 1 is a filter regular sequence by Corollary 3.4. Applying Remark 2.2 again, we get J = U 2 + a n R + a 1 R. Now let S = R/ 2. Then S is an FLC local ring of imension 2 an J / 2 = U a 1 S +a n S = U an S +a 1S for any integer n>0. Hence by Lemma 3.7, itfollows thath 1 S = 0. Therefore, R is a Cohen Macaulay ring by Lemma 3.6, since epth R 1. This completes the proof. REFERENCES [BH] W. Bruns an J. Herzog, Cohen Macaulay Rings, Cambrige Stuies in Avance Mathematics, Vol. 39, Cambrige Univ. Press Cambrige/New York/Port Chester/Syney, [CTS] N. T. Cuong, N. V. Trung, an P. Shenzel, Verallgemeinerte Cohen Macaulay-Mouln, Math. Nachr. 85 (1978), [FW] R. Feer an K. Watanabe, A characterization of F-regularity in terms of F-purity, in Commutative Algebra (M. Hochster, J. D. Sally, an C. Huneke, Es.), Math. Sc. Res. Inst. Publ. 15, pp , Springer-Verlag, Berlin/New York, [GN] S. Goto an Y. Nakamura, On the Finiteness of sup l R /, in preparation. [GY] S. Goto an K. Yamagishi, The Theory of Unconitione Strong -Sequences an Moules of Finite Local Cohomology, preprint, [HH] M. Hochster an C. Huneke, Tight closure, invariant theory, an the Brianco n Skoa theorem, J. Amer. Math. Soc. 3 (1990), [WY] K. Watanabe an K. Yoshia, Hilbert Kunz multiplicity an an inequality between multiplicity an colength, J. Algebra 230 (2000),