Uniform annihilators of local cohomology of extended Rees rings

Accepted Manuscript Uniform annihilators of local cohomology of extended Rees rings

Yi Qiu, Caijun Zhou

PII: DOI: Reference:

S0021-8693(17)30661-0 https://doi.org/10.1016/j.jalgebra.2017.12.007 YJABR 16491

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Journal of Algebra

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30 November 2016

Please cite this article in press as: Y. Qiu, C. Zhou, Uniform annihilators of local cohomology of extended Rees rings, J. Algebra (2018), https://doi.org/10.1016/j.jalgebra.2017.12.007

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Uniform annihilators of local cohomology of extended Rees rings Yi Qiu

Caijun Zhou

Department of Mathematics, Shanghai Normal University, Shanghai, 200234, China

Abstract The purpose of the paper is to study the annihilators of local cohomology of the extended Rees rings. We will prove that if (A, m) is a noetherian local ring with infinite residue field, and a is a uniform local cohomological annihilator of A, then there is an integer n such that (at)n is a uniform local cohomological annihilator of the extended Rees ring A[t, t−1 I] for any m-primary ideal I of A.

1

Introduction

Let A be a noetherian ring, I an ideal of A, and t an indeterminate over A. Consider A[t, t−1 ] as a graded ring in the natural way with deg(t) = 1. The extended Rees ring of A associated to I is the graded subring R = A[t, t−1 I] of the ring A[t, t−1 ]. If I = (a1 , a2 , · · · , ar ) then R[t, t−1 I] = R[t, t−1 a1 , · · · , t−1 ar ], so that R is also a noetherian ring. It is known that = dim(A) + 1 [cf. Ma].  dim(R) n n+1 I /I be the associated graded ring of A with respect to I. Let grI (A) = n≥0 One can show easily that R/tR  grI (A) and R/(1 − t)R  A, so that we can regard grI (A) as a deformation of the original ring A, with R as total space of the deformation, in the sense that the values t = 1 and 0 correspond to A and grI (A), respectively. Such deformation plays an important role in the theory of algebraic geometry [cf. Fu]. It is well known that if (A, m) is a d-dimensional Cohen-Macaulay (abbr. CM) local ring and I is an ideal generated by a system of parameters a1 , a2 , · · · , ad , then the extended Rees ring R is also a CM ring [cf, BH], and the result is not valid for all m-primary ideals. This means that there is a m-primary ideal I and a maximal ideal Q of A[t, t−1 I] such that HQi (A[t, t−1 I]) may be nonzero for some i (i < ht(Q)). In fact, such local cohomology modules are rarely finitely generated. In this paper, we will study the annihilators of these modules. Recalling that an element a ∈ A is said to be a uniform local cohomological annihilator of A, if it is not lying in any minimal prime ideal of A, and for any maximal ideal 1

m of A, a kills the i-th local cohomology module Him (A) for i < ht(m). It was proved in [Zh1] [Zh2] that if a locally equidimensional ring A satisfies one of the following conditions: (i) A is an excellent ring. (ii) A is a quotient ring of a CM ring of finite dimension. Then A has a uniform local cohomological annihilator. For more detailed information about this notion, one can find in [Sc], [Zh1] and [Zh2]. The main result of the paper is the following: Theorem 4.2 Let (A, m) be a d-dimensional noetherian local ring with infinite residue field. If A has a uniform local cohomological annihilator a, then there exists a positive integer n such that (at)n is a uniform local cohomological annihilator of the extended Rees ring A[t, t−1 I] for every m-primary ideal I of A.

2 Uniform local cohomological annihilators In this section, we recall and prove some facts about the uniform local cohomological annihilator which will be used in the paper. First of all, we prove a result which is an improvement of [Zh2, Lemma 3.5], where a is assumed to be a strong uniform local cohomological annihilator of A. Note that a uniform local cohomological annihilator a of A is said to be a strong uniform local cohomological annihilator of A if for any i (AP ) for i < ht(P). prime ideal P of A, a kills HPA P Lemma 2.1 Let A be a d-dimensional noetherian ring with a uniform local cohomological annihilator a. Let a1 , a2 , · · · , ar be a sequence of elements of A such that ht((a1 , a2 , · · · , ar )) = r. Then for any i, 1 ≤ i ≤ r, we have d

i−1 i−1 ) : ani i ) ⊆ ((an11 , an22 , · · · , ani−1 ) : ad ) ((an11 , an22 , · · · , ani−1

for all positive integers n1 , n2 , · · · , nr . Proof. To prove the conclusion, we may assume that A is a complete local ring with the unique maximal ideal m by [BH, Corollary 8.14]. Just as in the proof of [Zh1, Theorem 2.2], one conclude that a is also a strong uniform local cohomological annihilator of A. Thus the conclusion follows from [Zh2, Lemma 3.5]. The next result is a partial converse of Lemma 2.1, it is a particular case of the main result of [Sc] or [Zh1, Lemma 4.5]. We quote it here for later use. Lemma 2.2 Let (A, m) be a noetherian local ring of dimension d. Let a be an element of m and a1 , a2 , · · · , ad a system of parameters of A. If for each i (1 ≤ i ≤ d) i−1 i−1 ) : ani i ) ⊆ ((an11 , an22 , · · · , ani−1 ) : a) ((an11 , an22 , · · · , ani−1

hold for all positive integers n1 , n2 , · · · , nd , then ad Hmi (A) = 0 for i < d. In the rest of the paper, we need the following result which is a slight improvement of the main result of [LS]. As it was written in Chinese, we give it a shorter proof for readers’ convenience. 2

Proposition 2.3 Let A be a d-dimensional noetherian ring with a uniform local cohod mological annihilator a. Then ad (d+1) is a uniform local cohomological annihilator of the polynomial ring A[X] in one variable X. Proof. Clearly, every minimal prime ideal of A[X] is of the form PA[X], where P is a minimal prime ideal of A. So a does not lie in any minimal prime ideal of A[X], and d i (A[X]) = 0 for it suffices to show that for any maximal ideal M of A[X], ad (d+1) H M i < ht(M). Put m = M ∩ A and set r = ht(m). Choose elements a1 , a2 , · · · , ar in m such that ht((a1 , · · · , ar )) = r. By Lemma 2.1, for 1 ≤ i ≤ r, we have d

i−1 i−1 ((an11 , an22 , · · · , ani−1 ) :A ani i ) ⊆ ((an11 , an22 , · · · , ani−1 ) :A ad )

for all positive integers n1 , n2 , · · · , nr . Since A[X] is flat over A, it shows d

i−1 i−1 ((an11 , an22 , · · · , ani−1 ) :A[X] ani i ) ⊆ ((an11 , an22 , · · · , ani−1 )A[X] :A[X] ad )

and thus d

i−1 i−1 ) :A[X]M ani i ) ⊆ ((an11 , an22 , · · · , ani−1 ) :A[X]M ad ) ((an11 , an22 , · · · , ani−1

(2.3)

for 1 ≤ i ≤ r and for all positive integers n1 , n2 , · · · , nr . Replacing A by Am , we may assume A is a local ring with the maximal ideal m. Note that M/(mA[X]) is a prime ideal of (A/m)[X]. There is a monic polynomial f (X) ∈ M such that M = mA[X] + f (X)A[X]. It is clear that ht(M) = r + 1, (a1 , a2 , · · · , ar , f (X))A[X] M is a MA[X] M -primary ideal of A[X] M and ((an11 , an22 , · · · , anr r ) :A[X]M f (X)nr+1 ) = (an11 , an22 , · · · , anr r )A[X] M

(2.4)

for all positive integers n1 , n2 , · · · , nr+1 . Hence by (2.3), (2.4) and Lemma 2.2, we have ad

d

(r+1)

i H MA[X] (A[X] M ) = 0 M

i i (A[X]) = H MA[X] (A[X] M ). It shows ad for i < r + 1. Note that H M M for i < r + 1 , and the conclusion follows from r + 1 ≤ d + 1.

3

d

(r+1)

i HM (A[X]) = 0

Extended Rees rings of parameter ideals

In this section, we discuss the uniform local cohomological annihilators of extended Rees rings of parameter ideals. The result indicates that if a is a uniform local cohomological annihilator of a ring local ring A, then a fixed power of a is also a uniform local cohomological annihilator for the extended Rees ring of every parameter ideal of A. 3

We begin with some notations which will be used in the rest of the section. Let (A, m) be a noetherian local ring and I an ideal of A generated by a system of parameters a1 , a2 , · · · , ad . We denote the extended Rees ring A[t, t−1 I] by R, and the polynomial ring A[t, X1 , X2 , · · · , Xd ] in variables t, X1 , X2 , · · · , Xd by T . As in [Da], we consider the natural surjective morphism of A[t]-algebras φ : T → R which maps each Xi to t−1 ai for 1 ≤ i ≤ d. Let J be the kernel of φ, and K be the ideal of T generated by d elements tX1 − a1 , · · · , tXd − ad . It is clear that K is contained in J and R = T/J. The following result is observed by Davis [Da, 1.a and Proposition 1]. Lemma 3.1 Let J, K be as above. We have: (i) There is an integer n such that tn J ⊆ K. (ii) Radical J = radical K. Moreover, we have: Lemma 3.2 ht(J) = ht(K) = d. Proof. By Lemma 3.1 (ii). It suffices to prove ht(K) = d. Let P be an arbitrary minimal prime of K. Since t is not a zero-divisor of R, it implies that t does not lie in any minimal prime of J , and thus t  P by Lemma 3.1 (ii) again. Hence ht(P) = ht(PA[t, t−1 , X1 , · · · , Xd ]). Obviously, tX1 − a1 , · · · , tXd − ad is a regular sequence in A[t, t−1 , X1 , · · · , Xd ]. Hence ht(PA[t, t−1 , X1 , · · · , Xd ]) = d and the conclusion follows immediately. Now, we turn to the main result of the section. Theorem 3.3 Let (A, m) be a d-dimensional noetherian ring. If A has a uniform local cohomological annihilator a, then there exists a positive integer N, depending only on d, such that aN is a uniform local cohomological annihilator of the extended Rees ring A[t, t−1 a1 , · · · , t−1 ad ] for any system of parameters a1 , a2 , · · · , ad . Proof. Let R, T , J and K be as above. Put S = T/K. By means of Proposition 2.3 repeatedly, we conclude that there exists an integer n, depending only on d, such that an is a uniform local cohomological annihilator of T . Note that dim(T ) = 2d + 1. For any elements f1 , f2 , · · · , f s of T with ht( f1 , f2 , · · · , f s ) = s n

n

n

j−1 j−1 (( f1n1 , f2n2 , · · · , f j−1 ) : f j j ) ⊆ (( f1n1 , f2n2 , · · · , f j−1 ) : an(2d+1)

2d+1

)

(3.1)

hold for 1 ≤ j ≤ s and all positive integers n1 , n2 , · · · , n s . Let M be an arbitrary fixed maximal ideal of S with ht(M) = r. By Lemma 3.2, ht(K) = d, one can choose elements f1 , f2 , · · · , fr ∈ A[t, X1 , · · · , Xd ] such that ht(tX1 − a1 , · · · , tXd − ad , f1 , · · · , fr ) = d + r, the image xi of fi in S lies in M for each i and ht(x1 , · · · , xr ) = r. By (3.1), we have for each j, 1 ≤ j ≤ r, n

n

n

2d+1

j−1 j−1 ((x1n1 , x2n2 , · · · , x j−1 ) :S x j j ) ⊆ ((x1n1 , x2n2 , · · · , x j−1 ) :S an(2d+1)

4

)

hold for all positive integers n1 , n2 , · · · , nr . Localizing S at M, we obtain in the local ring S M that n

n

n

j−1 j−1 ) :S M x j j ) ⊆ ((x1n1 , x2n2 , · · · , x j−1 ) :S M an(2d+1) ((x1n1 , x2n2 , · · · , x j−1

2d+1

)

hold for all positive integers n1 , n2 , · · · , nr . Hence it follows from Lemma 2.2 arn(2d+1)

2d+1

i H MS (S M ) = arn(2d+1) M

2d+1

i HM (S ) = 0

(3.2)

for i < r. On the other hand, t is a non zero-divisor of R = T/J. By Lemma 3.1 (ii), t does not lie in any minimal prime of K, so ht(tX1 − a1 , · · · , tXd − ad , t) = d + 1. It follows from (3.1) that 2d+1

((tX1 − a1 , · · · , tXd − ad ) : tnd+1 ) ⊆ ((tX1 − a1 , · · · , tXd − ad ) : an(2d+1)

)

for all positive integers nd+1 . Hence by Lemma 3.1 (i), it shows 2d+1

an(2d+1)

J ⊆ K.

(3.3)

Let us consider the following natural short exact sequence of S -modules 0 → J/K → S → R → 0. For the maximal ideal M of S , we have the long exact sequences of local cohomology of S -modules i i i i+1 (J/K) → H M (S ) → H M (R) → H M (J/K) → · · · . · · · → HM

(3.4)

2d+1

i By (3.2) and (3.3), we conclude easily that a(r+1)n(2d+1) H M (R) = 0 for i < r. Since r ≤ d + 1, we have proved that for any maximal ideal M of S , 2d+1

a(d+2)n(2d+1)

i HM (R) = 0

(3.5)

for i < ht(M). Now, If M  is an arbitrary maximal ideal of R, then there exists a maximal ideal M of S such that M  = MR. It follows from Lemma 3.1 (ii), ht(M  ) = ht(MR).

(3.6)

i i Moreover, we have H M  (R) = H M (R) by [Sh]. So it follows from (3.5) and (3.6) that (d+2)n(2d+1)2d+1 i H M (R) = 0 for i < ht(M  ). Put N = (d + 2)n(2d + 1)2d+1 . Clearly, N is a only dependent on d, and i aN H M  (R) = 0

for i < ht(M  ) and for all maximal ideal M  of R. Note that each minimal prime ideal of R has the form pA[t, t−1 ] ∩ R, where p is a minimal prime of A [cf, Ma, 15.4]. It shows aN is not contained in any minimal prime of R, and consequently aN is a uniform local cohomological annihilator of R. This ends the proof of the theorem. 5

4 Extended Rees rings of m-primary ideals Let (A, m) be a d-dimensional noetherian local ring with infinite residue field, and a a uniform local cohomological annihilator A. We consider the question whether there exists an element of A[t] such that it is a uniform local cohomological annihilator of A[t, t−1 I] for every m-primary ideal I of A. For an ideal I of A, Northcott and Rees [NR] showed there are elements a1 , a2 , · · · , ad contained in I such that the ideal (a1 , a2 , · · · , ad ) is a reduction ideal of I, that is, there is an integer n, depending on I, such that I n+1 = (a1 , a2 , · · · , ad )I n . In particular, if I is a m-primary ideal of A, the elements a1 , a2 , · · · , ad form a system of parameters of A. We will proceed with by relating the existence of the uniform local cohomological annihilator of A[t, t−1 I] to that of the ring A[t, t−1 a1 , · · · , t−1 ad ]. Note that, if A is a reduced excellent local ring, Huneke [Hu] proved that there is an integer k, such that for n ≥ k I n ⊆ J n−k hold for all ideals I and all reduction ideals J of I. Recently, Zhou noted that the conclusion is valid for all noetherian local rings. We quote it as a lemma. Lemma 4.1 [Zh3, Corollary 3.7] Let (A, m) be a noetherian local ring. Then there is an integer k such that for any ideal I of A and any reduction ideal J of I, I n ⊆ J n−k for all n ≥ k. The main result of the paper is the following which is unknown even in the case that A is a local CM ring. Theorem 4.2 Let (A, m) be a d-dimensional noetherian local ring with infinite residue field, and a be a uniform local cohomological annihilator A. Then there exists a positive integer n such that an tn is a uniform local cohomological annihilator of A[t, t−1 I] for every m-primary ideal I of A. Proof. Let I be an arbitrary fixed m-primary ideal of A. As the residue field of A is infinite, it follows from Northcott and Rees [NR] that there is a system of parameters a1 , a2 , · · · , ad of A such that the ideal (a1 , a2 , · · · , ad ) is a reduction of I. Set R = A[t, t−1 a1 , · · · , t−1 ad ], S = A[t, t−1 I], and L = S /R. It is easy to see that S is a finite R module. In particular, S is integral over R. By Lemma 4.1, there exists an integer k, depending only on A, such that tk L = 0. Consider the short exact sequence of finite R-modules 0 → R → S → L → 0. 6

(4.1)

For an arbitrary fixed maximal ideal Q of S, put P = Q ∩ R. It is clear that P is also a maximal ideal of R. We have the long exact sequence of local cohomology modules · · · → HPj (R) → HPj (S ) → HPj (L) → HPj+1 (R) → · · · . By Theorem 3.3, there is an integer n, not depending on P, such that an HPj (R) = 0 for j < ht(P). Hence by (4.1), we have an tk HPj (S ) = 0 for j < ht(P). Replacing n by n+k if necessary, we may assume an tn HPj (S ) = 0 for j < ht(P). Hence by the properties of local cohomology [Sh], we have j (S ) = 0 an tn HPS

(4.2)

for j < ht(P). Suppose that PS = q1 ∩ q2 ∩ · · · ∩ qr is an irredundant primary decomposition of the ideal PS of S with AssS (S /qi ) = {Qi } for 1 ≤ i ≤ r. We know {Q1 , Q2 , · · · , Qr } is the set of maximal ideals of S lying over P [cf, Ma, Theorem 9.3]. Without loss of generality , we may assume that Q1 = Q. Note that for j ≥ 0, j (S ) = HQj 1 ∩Q2 ···∩Qr (S ). HPS Since each Qi is a maximal ideal, we conclude by Mayer-Vietoris sequence of local cohomology that for j ≥ 0, HQj 1 ∩Q2 ···∩Qr (S ) HQj 1 (S ) ⊕ HQj 2 (S ) ⊕ · · · ⊕ HQj r (S ). Hence by (4.2), we have an tn HQj i (S ) = 0

(4.3)

for j < ht(P). To end the proof of the theorem, we need the following claim. Claim. ht(Qi ) = ht(P) for 1 ≤ i ≤ r. Proof. For each i, if Q0 is a minimal prime ideal of S with Q0 ⊂ Qi , then there is a prime ideal p of A such that Q0 = pA[t, t−1 ] ∩ S , and P0 = Q0 ∩ R is also a minimal prime of R [Ma, 15.4]. By Theorem 3.3, R has a uniform local cohomological annihilator. So it follows from [Zh1, Theorem 2.1] that R is locally equidimensional. It implies ht(P/P0 ) = ht(P).

(4.4)

By [Zh1, Theorem 2.1] again, R is universally catenary. Since S /Q0 is integral over R/P0 , it follows from the dimension formula between R/P0 and S /Q0 that ht(Qi /Q0 ) = ht(P/P0 ) [cf, Ma, Theorem 15.6]. Hence by the arbitrary choice of Q0 and (4.4), we conclude that ht(Qi ) = ht(P) for 1 ≤ i ≤ r, and this proves the claim.

7

Now, we continue the proof of the theorem. By the claim and (4.3), we have shown that for any maximal ideal Q of S an tn HQj (S ) = 0 for j ≤ ht(Q). As every minimal prime of S must have the form pA[t, t−1 ] ∩ S , where p is a minimal prime of A [cf, Ma, 15.4]. we assert that an tn does not lie in any minimal prime ideal of S , Thus an tn is a uniform local cohomological annihilator of S , and the proof is complete. Acknowledgment The authors are grateful for the referee for his or her helpful suggestions and comments. References [BH] Bruns, W., Herzog, J.: Cohen-Macaulay Rings. Cambridge Univ. Press, Cambridge, (1993) [Da] Davis, E. D.: Ideals of principal classes, R-sequences and a certain monoidal transformation. Pacific J. Math. 20, 197-205 (1967) [Fu] Fulton, W.: Intersection Theory. Springer-Verlag, New York (1984) [Gr] Grothendieck, A.: Local Cohomology. Lect. Notes Math. 41, Springer-Verlag, New York (1963) [Hu] Huneke, C.: Uniform bounds in noetherian rings. Invent. Math. 107, 203-223 (1992) [LS] Liu, G., Song, C.: Uniform annihilators of local cohomology and polynomial extension(in Chinese). J. Shanghai Normal University 35 52-55 (2006). [Ma] Matsumura, H.: Commutative Ring Theory, Cambridge Univ. Press, New York, (1986) [NR] Northcott, D. G., Rees, D.: Reductions of ideals in local rings. Proc. Cambridge Phil. Soc. 50, 145-158 (1954) [Sc] Schenzel, P.: Cohomological annihilators. Math. Proc. Camb. Phil. Soc. 91, 345-350 (1982) [Sh1] Sharp, R.: Local cohomology theory in commutative algebra. Q. J. Math. Oxford 21, 425-434 (1970) [Zh1] Zhou, C.: Uniform annihilators of local cohomology. J. Algebra 305, 585-602 (2006) [Zh2] Zhou, C.: Uniform annihilators of local cohomology of excellent rings. J. Algebra 315, 286-300 (2007) [Zh3] Zhou, C.: Integral dimension of a noetherian ring, preprint, arXiv:1603.00045.

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