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Sequentially Cohen-Macaulay rings and modules

Truong Le HOANG VAST - 11034

Researcher,Department of Algebra,Institute of Mathematics, Vietnam Academy of Science and Technology

Japanese Advisor : Shiro GOTOProfessor, Meiji University

The purpose of my thesis is to study sequentially Cohen-Macaulay rings and modules. The notion of sequentially Cohen-Macaulay ring and module was introduced by R. P. Stanley [St] and has been intensively explored by D. T. Cuong and N. T. Cuong [CC1]. In this thesis I shall continue these researches, exploring the structure of certain special, say good and distinguished, systems of parameters.

My thesis consits of three chapters. Let me explain the contents of each chapter and the main results as well, fixing the notation and terminology which I maintain through my thesis.

In Chapter 1, let (R, ) be a commutative Noetherian local ring and M a finitely generated R-module of dimension d. We consider a finite filtration of submodules of M, : M0 ⊂ M1 ⊂ . . . ⊂ Mt = M such that dim M0 < dim M1 < . . . < dim Mt = dim M. Such a filtration is said to satisfy the dimension condition. A filtration : H 0 (M) = D0 ⊂ D1 ⊂ . . . ⊂ Dt = M of submodules of M is said to be a dimension filtration, if Di−1 is the largest submodule of Di with dim Di−1 < dim Di for all i = t, t − 1, . . . , 1. If Di / Di−1 is Cohen-Macaulay for all 1 ≤ i ≤ t, M is called a sequentially Cohen-Macaulay module. Let x = x1, . . . , xd be a system of parameters of M. Then x is called a good system of parameters with respect to if Mi ∩ (xdi+1, . . . , xd)M = 0 for i = 0, 1, . . . , t − 1, where di = dim Mi. A good system of parameters with respect to the dimension filtration is simply called a good system of parameters of M. Set

where e(x1, . . . , xdi ; Mi) is the Serre multiplicity of Mi relative to x1, . . . , xdi

and ℓR(N) denotes, for an R-module N, the length of N. Denote (n) = x(n) = xn1

1 , . . . , xnd

d for any d-tuple n = (n1, . . . , nd) of positive integers. Notice that I , M (x) = ℓR(M/xM) − arith-deg((x); M). It is shown in [CC1] that I , M (x) is non-negative and I , M (x(n)) is non-decreasing as a function in n1, . . . , nd. In fact, I , M (x(n)) is not a polynomial in n1, . . . , nd in general. However, it can be seen easily that this function is bounded above by a polynomial. In Chapter 1 we study the least degree of the polynomials bounding above I , M (x(n)) and show that this degree is independent of the choices of good systems of parameters with respect to and

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compute this degree in the case where is the dimension filtration of M. This is the content of the following theorem.

Theorem. Let : M0 ⊂ M1 ⊂ . . . ⊂ Mt = M be a filtration of submodules of M satisfying the dimension condition and x = x1, . . . , xd a good system of parameters with respect to . Then

(1) (Theorem 1.1.1) The least degree of all polynomials in n1, . . . , nd bounding above the function I , M (x(n)) is independent of the choice of x, and denoted by p (M).

(2) (Theorem 1.1.2) Let : D0 ⊂ D1 ⊂ . . . ⊂ Dt = M be the dimension filtration of M. Assume that R is a quotient of a Cohen-Macaulay ring. Then we have

where V (M) is the non-sequentially Cohen-Macaulay locus of M and p(M) is the polynomial type of M [C].

For each integers n 1, we set

We say that the system x of parameters has the property of parametric decomposition, if the equality nM = ∩α∈Λd, n

(α)M holds true for all n 1. The main purpose of the final two sections of Chapter 1 is to study the question of when a given system of parameters of M has the property of parametric decomposition. Now, restrict our interest in the above question to the set of all good systems of parameters of M. It turns out that the property of parametric decomposition of a good system of parameters can be characterized by the sequentially Cohen-Macaulayness of the module. The following theorem is the main result of Chapter 1.

Theorem (Theorem 1.1.3). Let R be a commutative Noetherian local ring with the maximal ideal and M a finitely generated R-module with dim M = d > 0. The following statements are equivalent:(1) M is a sequentially Cohen-Macaulay module.(2) Every good system of parameters of M has the property of parametric

decomposition.(3) There exists a good system of parameters of M having the property of

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parametric decomposition.

Let I be an -primary ideal of R. It is well-known that there exists a polynomial pI (n) of degree d with rational coefficients, called the Hilbert-Samuel polynomial, such that ℓR(M/I n+1M) = pI (n) for all integers n large enough. Then, there are integers ei(I, M) such that

These integers ei(I, M) are called the Hilbert coefficients of M with respect to I. In particular, the leading coefficient e0(I, M) is called the multiplicity of M with respect to I and e1(I, M) is called by Vasconcelos the Chern number of I with respect to M. At the conference in Yokohama 2008, W. V. Vasconcelos [V4] posed the following conjecture:

The Vanishing Conjecture: Assume that R is an unmixed, that is dim(R̂/P) = dim R for all P ∈ Ass R̂, where R̂ is the -adic completion of R. Then R is a Cohen-Macaulay local ring if and only if e1( , R) = 0 for some parameter ideal of R.

Recently, this conjecture has been settled by L. Ghezzi, S. Goto, J.-Y. Hong. K. Ozeki, T. T. Phuong, W. V. Vasconcelos in [GGHOPV]. The aim of Chapter 2 is to give characterizations of a sequentially Cohen-Macaulay module in term of its Hilbert coefficients and Hilbert-Samuel functions with respect to good parameter ideals. Recall that for all i, the i-th arithmetic degree of M with respect to I is defined by arith-degi(I, M) = multM( )e0(I ,R / ), ([BM], [V2], [V3])

Theorem (Theorem 2.2.19 and 1.3.7). Assume that R is a homomorphic image of a Cohen-Macaulay local ring. Let Λ(M) = {r ∈ | there is a submodule N of M such that dim N = r}. Then the following statements are equivalent:

(1) M is a sequentially Cohen-Macaulay R-module.

(2) For any good parameter ideal of M, it holds

for all n 0.

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(3) There exists a good parameter ideal of M such that

for all n 0.

(4) For all distinguished parameter ideals of M and j = 0, . . . , d, we have

(5) For all distinguished parameter ideals of M and j ∈ Λ (M), we have

(6) For some distinguished parameter ideal of M and for all j ∈ Λ (M), we have

We say that an R-submodule N of M is irreducible, if N is not written as the intersection of two larger R-submodules of M. Every R-submodule N of M can be expressed as an irredundant intersection of irreducible R-submodules of M and the number of irreducible R-submodules appearing in such an expression depends only on N and is independent of the expression. Let us call, for each parameter ideal of M, the number N( ;M) of irreducible R-submodules of M that appear in an irredundant irreducible decomposition of M the index of reducibility of M with respect to . We can ask whether sequentially Cohen-Macaulay modules have eventual constant index of reducibility for parameter ideals. This is, unfortunately, not true in general, as Rogers [R, Example 4.3] gave counter-examples. However, once we restrict our attention to good parameter ideals of M, the answer is affirmative. The main result of Chapter 2 is stated as follows.

Theorem (Theorem 2.1.1). Let R be a Noetherian local ring with maximal ideal and M a finitely generated R-module of dimension d > 0. If M is a sequentially Cohen-Macaulay R-module, then there is an integer n ≫ 0 such that for every distinguished parameter ideal of M contained in n, one has the equality

Hence the index of reducibility of M with respect to parameter ideals generated by distinguished systems of parameters is eventually constant.

Unless otherwise specified, throughout Chapter 3 let R be a commutative Noetherian local ring with maximal ideal and d = dim R > 0.

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Problem A (Question 3.1.1). Let R be a sequentially Cohen-Macaulay ring. Assume that dim R ≥ 2, or that dim R = 1 and e(R) > 1. Then does there exist an integer n > 0 such that I 2 = I for every parameter ideal contained in n, where I = :R ?This is one of the targets of my research.

To state the results of Chapter 3, f irst of all let us f ix our notation and terminology. For a while, let R be a (not necessarily local) commutative Noetherian ring and let M (≠ (0)) be a finitely generated R-module of finite Krull dimension. Then a filtration

of R-submodules of M is called the dimension filtration of M, if for all 1 ≤ i ≤ ℓ Di−1 is the largest R-submodule of Di with dimR Di−1 < dimR Di, where dimR(0) = −∞ for convention. Here we notice that our notion of dimension filtration is based on [GHS] and slightly different from that of [CC1, Sch]. We say that M is a sequentially Cohen-Macaulay R-module, if Ci = Di/Di−1 is a Cohen-Macaulay R-module (necessarily with dimR Ci = dimR Di) for all 1 ≤ i ≤ ℓ ([Sch, St]). Hence M is a sequentially Cohen–Macaulay R–module with ℓ = 1 if and only if M is a Cohen-Macaulay R-module with dim R/ = dimR M for every ∈ AssRM. We say that R is a sequentially Cohen-Macaulay ring, if dim R < ∞ and R is a sequentially Cohen-Macaulay module over itself.

We now assume that R is a local ring with maximal ideal and let M be a finitely generated R-module with d = dimRM ≥ 1 and = {Di}0 ≤ i ≤ ℓ the dimension filtration. Let x = x1, x2, . . . , xd be a system of parameters of M. Remember that x is said to be good (resp. distinguished), if

for all 1 ≤ i ≤ ℓ , where di = dimR Di. A parameter ideal of M is called good (resp. distinguished), if there exists a good (resp. distinguished) system x1, x2, . . . , xd of parameters of M such that = (x1, x2, . . . , xd). Therefore, if M is a Cohen-Macaulay R-module, every parameter ideal of M is good. Notice that good parameter ideals are distinguished in due course. The converse is also true, if M is a sequentially Cohen-Macaulay R-module ([CC1]).

Good system of parameters exist ([CC1, Lemma 2.4]) and if x1, x2, . . . , xd is a good system of parameters of M, then xn1

1 , xn22 , . . . , x

ndd is also a good system of

parameters of M for all integers nj ≥ 1.Let I (≠ R) be an ideal of R. We put

(here t denotes an indeterminate over R), which we call, respectively, the Rees algebra and the extended Rees algebra of I. Let grI (R) = '(I)/t−1 '(I) and call it the

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associated graded ring of I. We shall usually identify

For each ideal D of R, the family = {I n ∩ D}n ∈ Z forms an I-stable filtration of D, for which the graded module gr( ) = n≥0[I

n ∩ D]/[I n+1 ∩ D] is a graded ideal of grI (R) and the graded module ( ) = n≥0[I

n ∩ D] (resp. '( ) = n∈Z[I n ∩ D]) is a graded ideal of (I) (resp. '(I)).

With this notation the main results of Chapter 3 are summarized into the following, which gives a complete generalization of the results in the Cohen-Macaulay case to those of sequentially Cohen-Macaulay rings.

Theorem (Theorem 3.1.2). Let R be a sequentially Cohen-Macaulay local ring with d = dim R ≥ 1 and let = {Di}0≤i≤ℓ be the dimension filtration of R. Let be a good parameter ideal of R and assume that

We put I = :R and S = R/Dℓ−1. Then the following conditions are equivalent.(1) I 2 = I.

(2) The parameter ideal S of S is not integrally closed in S.

When this is the case, the following assertions hold true.

(a) The associated graded ring grI (R) = n≥0 In/I n+1 of I is a sequentially

Cohen-Macaulay ring with {gr({I n ∩ Di}n∈ )}0≤i≤ℓ the dimension filtration.

(b) The extended Rees algebra '(I) = n∈ I n of I is a sequentially Cohen-Macaulay ring.

(c) Let Λ (R) = {di | 1 ≤ i ≤ ℓ} where di = dimR Di.Then

H j (grI (R)) = (0),if j ∉ Λ (R) and

if 1 ≤ i ≤ ℓ and di > 0, where = ・ grI (R) + [grI (R)]+ denotes the graded maximal ideal of grI (R).

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We therefore have the following.

(d) The Rees algebra (I ) = n≥0 In of I is a sequentially Cohen-Macaulay

ring, if d1 ≥ 2, or if d1 = 0 but d2 ≥ 2.

Combining Theorem 3.1.2 with Theorem 2.1.1, we get the following answer to Question 3.1.1.

Corollary (Corollary 3.1.3). Let R be a sequentially Cohen-Macaulay local ring with maximal ideal . Assume that dimR ≥ 2, or that dimR = 1 and e(R) > 1. Then there is an integer n ≫ 0 such that for every good parameter ideal of R contained in n, one has the equality I 2 = I, where I = : R , so that assertions (a), (b), (c), and (d) of above Theorem hold true for the ideal I.

参考文献[BM] D. Bayer and D. Mumford, What can be computed on algebraic

geometry?, Computational Algebraic Geometry and Commutative algebra, Proceedings. Cortona 1991(D. Eisenbud and L. Robbiano Eds), Cambridge University Press (1993), pp 1-48.

[CC1] D. T. Cuong and N. T. Cuong, On sequentially Cohen-Macaulay modules, Kodai Math. J., 30 (2007), pp. 409-428.

[C] N. T. Cuong, On the least degree of polynomials bounding above the differences between lengths and multiplicities of certain system of parameters in local rings, Nagoya Math. J. 125 (1992), pp. 105-114.

[GGHOPV] L. Ghezzi, S. Goto, J.-Y. Hong. K. Ozeki, T. T. Phuong, and W. V. Vasconcelos, The first Hilbert coefficients of parameter ideals, J. London Math. Soc. (2) 81 (2010), pp. 679-695.

[GHS] S. Goto, Y. Horiuchi, and H. Sakurai, Sequentially Cohen-Macaulayness versus parametric decomposition of powers of parameter ideals, J. Commutative Alg., 2 (2010), pp. 37-54.

[R] M. Rogers, The index of reducibility for parameter ideals in low dimension, J. Algebra, 278 (2004), pp. 571-584.

[Sch] P. Schenzel, On the dimension filtration and Cohen-Macaulay filtered modules, In. Proc. of the Ferrara meeting in honour of Mario Fiorentini, University of Antwerp Wilrijk, Belgium, (1998), pp. 245-264.

[St] R. P. Stanley, Combinatorics and Commutative Algebra, Second Ed., Birkhäuser, Boston, Basel, Stuttgart, 1996.

[V2] W. V. Vasconcelos, The degrees of graded modules, Lecture Notes in Summer School on Commutative Algebra, vol. 2, pp. 141-196, Centre

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de Recerca Matematica, Bellaterra (Spain), 1996.[V3] W. V. Vasconcelos, Computational Methods in Commutative algebra

and Algebraic Geometry, Springer Verlag, Berlin- Heidelderg-New York, 1998

[V4] W. V. Vasconcelos, The Chern coefficients of local rings, Michigan Math. J., 57 (2008), pp. 725-743.