Normal Sally modules of rank one

The Hilbert coefficients of the normal filtration give important geometric information on the base ring like the pseudo-rationality. The Sally module was introduced by W.V. Vasconcelos and it is useful to connect the Hilbert coefficients to the homological properties of the associated graded module of a Noetherian filtration. In this paper we give a complete structure of the Sally module in the case the second normal Hilbert coefficient attains almost minimal value in an analytically unramified Cohen-Macaulay local ring. As a consequence, in this case we present a complete description of the Hilbert function of the associated graded ring of the normal filtration. A deep analysis of the vanishing of the third Hilbert coefficient has been necessary. This study is related to a long-standing conjecture stated by S. Itoh. 2010 Mathematics Subject Classification. 13D40, 13A30, 13H10. Proposition 2.4. Let d ≥ 2. Then the graded T -module C (2) J ({I n } n∈Z ) satisfies the Serre's property (S 2 ) as a T /Ann T (C (2) J ({I n } n∈Z ))-module. 3. The structure of the Sally module when e 3 (I) = 0In this section we prove the first main result of this paper (Theorem 3.6). We set C = C J (I) = C(2) J ({I n } n∈Z ). In the following theorem we recall few results on the vanishing of local cohomology modules from [11] (see also [6]). From now onwards we set M ′ = (t −1 , m, It)R ′ (I) for the graded maximal ideal of R ′ (I) and N ′ = ItR ′ (I).

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“…We recall that, by using [14, Proposition 2.11] and [20], we have the following interesting result. Proposition 13]) Suppose that d ≥ 2.…”

Section: Notation and Preliminariesmentioning

confidence: 99%

“…Recently A. Corso, C. Polini and M. E. Rossi showed that if g s (I) = 1 holds true, then depth G(I) ≥ d − 1 (see CPR16). This case was also explored by T. T. Phuong [20] when R is a generalized Cohen-Macaulay ring.…”

Section: Introductionmentioning

confidence: 99%

On the structure of the Sally module and the second normal Hilbert coefficient

Masuti

Ozeki

Rossi

et al. 2020

Proc. Amer. Math. Soc.

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“…The proof is discussed in [6,Lemma 2.3] for I = {I n } and in [17,Lemma 2.2] for I = {I n }. We provide an analogous proof for general case for the convenience of reader.…”

Section: Preliminariesmentioning

confidence: 99%

Buchsbaumness of the associated graded rings of filtration

Saloni

2020

J. Algebra Appl.

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Let [Formula: see text] be a Noetherian local ring of dimension [Formula: see text] and [Formula: see text] an [Formula: see text]-primary ideal of [Formula: see text]. In this paper, we discuss a sufficient condition, for the Buchsbaumness of the local ring [Formula: see text] to be passed onto the associated graded ring of filtration. Let [Formula: see text] denote an [Formula: see text]-good filtration. We prove that if [Formula: see text] is Buchsbaum and the [Formula: see text] -invariant, [Formula: see text] and [Formula: see text], coincide then the associated graded ring [Formula: see text] is Buchsbaum. As an application of our result, we indicate an alternative proof of a conjecture, of Corso on certain boundary conditions for Hilbert coefficients.

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“…Recently [CPR16] showed that if the equality e 1 (I) = e 0 (I) − ℓ R (R/I) + 1 holds true, then depth G(I) ≥ d −1 (see also Corollary 4.3). This equality was explored by Phuong [Phu15] in the case R is generalized Cohen-Macaulay.…”

Section: Introductionmentioning

confidence: 99%

A filtration of the Sally module and the first normal Hilbert coefficient

2021

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The Sally module of an ideal is an important tool to interplay between Hilbert coefficients and the properties of the associated graded ring. In this paper we give new insights on the structure of the Sally module. We apply these results characterizing the almost minimal value of the first Hilbert coefficient in the case of the normal filtration in an analytically unramified Cohen-Macaulay local ring. 1 2 MASUTI, OZEKI, AND ROSSI e i (I), called the normal Hilbert coefficients of I, such that for n ≫ 0Here ℓ R (N) denotes, for an R-module N, the length of N. Since R/m is infinite there exists a minimal reduction J = (a 1 , . . . , a d ) of I and, under our assumptions, there exists a positive integer r such that I n+1 = JI n for n ≥ r. We set r J (I) := min{r ≥ 0 | I n+1 = JI n for all n ≥ r} the reduction number of I.We recall that e 1 (I) ≥ 0, but the bound can be more precise. It is well known thatMoreover, the equality holds true if and only if I n+1 = J n I for every n ≥ 0 and for every minimal reduction J of I (that is r J (I) ≤ 1) ([Nor60,Hun87,Ito92,Ooi87]). In this case the normal associated graded ring G(I) := ⊕ n≥0 I n /I n+1 of I is Cohen-Macaulay. Recently [CPR16] showed that if the equality e 1 (I) = e 0 (I) − ℓ R (R/I) + 1 holds true, then depth G(I) ≥ d −1 (see also Corollary 4.3). This equality was explored by Phuong [Phu15] in the case R is generalized Cohen-Macaulay. By [EV91, GR98, Ito92, HM97] it is known thatand the equality holds true if and only if I n+1 = J n−1 I 2 for every n ≥ 1 (that is r J (I) ≤ 2). In this case the normal associated graded ring G(I) of I is Cohen-Macaulay (see also Corollary 2.9). We notice that ℓ R (I 2 /JI) does not depend on a minimal reduction J of I (see for instance [RV10]). Thus the ideals I with e 1 (I) = e 0 (I) − ℓ R (R/I) + ℓ R (I 2 /JI) enjoy nice properties and it seems natural to ask when the equality e 1 (I) = e 0 (I) − ℓ R (R/I) + ℓ R (I 2 /JI) + 1 holds. The main purpose of this paper is to explore this equality which is the content of Section 3. We present in this case the structure of the Sally module and, in particular, we prove that depth G(I) ≥ d − 1 (Theorem 3.1). We remark that if I is integrally closed, the corresponding equality was studied in [OR16]. In [OR16] the authors proved that if the equality e 1 (I) = e 0 (I) − ℓ R (R/I) + ℓ R (I 2 /JI) + 1 holds true, then the depth of the associated graded ring G(I) := n≥0 I n /I n+1 can be any integer between 0 and d − 1 (see [OR16, Theorem 5.1]). This "bad" behavior motives our study in the case of the normal filtration proving that it enjoys nice properties as compared to the I-adic filtration.As the title outlines, an important tool in this paper is the Sally module introduced by W. V. Vasconcelos [Vas94]. The aim of the author was to define a module in between the associated graded ring and the Rees algebra taking care of important information coming from a minimal reduction. Actually, a more detailed information comes from the graded parts of a suitable filtration {C (i) } of the Sally modul...

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Normal Sally modules of rank one

Cited by 3 publications

References 11 publications

On the structure of the Sally module and the second normal Hilbert coefficient

On the structure of the Sally module and the second normal Hilbert coefficient

Buchsbaumness of the associated graded rings of filtration

A filtration of the Sally module and the first normal Hilbert coefficient

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