Cohen Macaulay Local Ring Research Articles

A sub-functor for Ext and Cohen–Macaulay associated graded modules with bounded multiplicity-II

Abstract Let $(A,\mathfrak{m})$ be a Cohen–Macaulay local ring, and then the notion of a $T$ -split sequence was introduced in the part-1 of this paper for the $\mathfrak{m}$ -adic filtration with the help of the numerical function $e^T_A$ . In this article, we explore the relation between Auslander–Reiten (AR)-sequences and $T$ -split sequences. For a Gorenstein ring $(A,\mathfrak{m})$ , we define a Hom-finite Krull–Remak–Schmidt category $\mathcal{D}_A$ as a quotient of the stable category $\underline{\mathrm{CM}}(A)$ . This category preserves isomorphism, that is, $M\cong N$ in $\mathcal{D}_A$ if and only if $M\cong N$ in $\underline{\mathrm{CM}}(A)$ .This article has two objectives: first objective is to extend the notion of $T$ -split sequences, and second objective is to explore the function $e^T_A$ and $T$ -split sequences. When $(A,\mathfrak{m})$ is an analytically unramified Cohen–Macaulay local ring and $I$ is an $\mathfrak{m}$ -primary ideal, then we extend the techniques in part-1 of this paper to the integral closure filtration with respect to $I$ and prove a version of Brauer–Thrall-II for a class of such rings.

Krull–Gabriel dimension of Cohen–Macaulay modules over hypersurfaces of countable Cohen-Macaulay representation type

We calculate the Krull–Gabriel dimension of the functor category of the (stable) category of maximal Cohen–Macaulay modules over complete Cohen-Macaulay local rings of countable Cohen-Macaulay representation type. We show that the Krull–Gabriel dimension is 0 if and only if the ring is of finite Cohen–Macaulay representation type and that is 2 if the ring is a hypersurface of countable but not finite Cohen–Macaulay representation type.

A note on Ratliff-Rush filtration, reduction number and postulation number of [formula omitted]-primary ideals

Let (R,m) be a Cohen-Macaulay local ring of dimension d≥2, and I an m-primary ideal. Let r(I) be the reduction number of I, n(I) the postulation number and ρ(I) the stability index of the Ratliff-Rush filtration with respect to I. We prove that for d=2, if n(I)=ρ(I)−1, then r(I)≤n(I)+2, and if n(I)≠ρ(I)−1, then r(I)≥n(I)+2. For d≥3, assuming I is integrally closed, depthgr(I)=d−2, and n(I)=−(d−3), we prove that r(I)≥n(I)+d. Our main result generalizes a result by Marley on the relation between the Hilbert-Samuel function and the Hilbert-Samuel polynomial by relaxing the condition on the depth of the associated graded ring to the good behavior of the Ratliff-Rush filtration with respect to I mod a superficial sequence. From this result, it follows that for Cohen-Macaulay local rings of dimension d≥2, if PI(k)=HI(k) for some k≥ρ(I), then PI(n)=HI(n) for all n≥k.

On local divisor class groups of complete intersections

Samuel conjectured in 1961 that a (Noetherian) local complete intersection ring that is a UFD in codimension at most three is itself a UFD. It is said that Grothendieck invented local cohomology to prove this fact. Following the philosophy that a UFD is nothing else than a Krull domain (that is, a normal domain, in the Noetherian case) with trivial divisor class group, we take a closer look at the Samuel–Grothendieck Theorem and prove the following generalization: Let A be a local Cohen–Macaulay ring.(1)A is a normal domain if and only if A is a normal domain in codimension at most 1.(2)Suppose that A is a normal domain and a complete intersection. Then the divisor class group of A is a subgroup of the projective limit of the divisor class groups of the localizations Ap, where p runs through all prime ideals of height at most 3 in A. We use this fact to describe for an integral Noetherian locally complete intersection scheme X the gap between the groups of Weil and Cartier divisors, generalizing in this case the classical result that these two concepts coincide if X is locally a UFD.

Semi-prime operation and superficial sequence

Let [Formula: see text] be a Cohen–Macaulay local ring with infinite residue field [Formula: see text] and [Formula: see text] be an [Formula: see text]-primary ideal of [Formula: see text]. Let [Formula: see text] be a semi-prime operation in the set of ideals of [Formula: see text] and [Formula: see text] be a [Formula: see text]-good filtration on [Formula: see text]. Let [Formula: see text] be an integer and [Formula: see text] for all large integers [Formula: see text]. In this paper, we give some relationships between [Formula: see text] and [Formula: see text]-superficial sequence. Thus, we show that if [Formula: see text] is a maximal [Formula: see text]-superficial sequence on [Formula: see text] and [Formula: see text], then it exists a positive integer [Formula: see text] such that [Formula: see text] for all [Formula: see text].

Generalized Loewy length of Cohen-Macaulay local and graded rings

We generalize a theorem of Ding relating the generalized Loewy length gℓℓ(R) and index of a one-dimensional Cohen-Macaulay local ring (R,m,k). Ding proved that if R is Gorenstein, the associated graded ring is Cohen-Macaulay, and k is infinite, then the generalized Loewy length and index of R are equal. However, if k is finite, equality may not hold. We prove that if the index of a one-dimensional Cohen-Macaulay local ring is finite and the associated graded ring has a homogeneous nonzerodivisor of degree t, then gℓℓ(R)≤index(R)+t−1. Next we prove that if R is a one-dimensional hypersurface ring with a witness to the generalized Loewy length that induces a regular initial form on the associated graded ring, then the generalized Loewy length achieves this upper bound. We then compute the generalized Loewy lengths of several families of examples of one-dimensional hypersurface rings over finite fields. Finally, we study a graded version of the generalized Loewy length and determine its value for numerical semigroup rings.

On André-Quillen homology and complete intersection ideals

We establish new characterizations of complete intersection ideals in Cohen-Macaulay local rings as a byproduct of our investigation on a certain André-Quillen homology vector space in connection to the vanishing of (finitely many) Ext modules over a suitable quasi-deformation. Along the way, by means of (co)homology vanishing, we also detect criteria for quasi-complete intersection ideals.

Reflexive modules over the endomorphism algebras of reflexive trace ideals

In the present paper we investigate reflexive modules over the endomorphism algebras of reflexive trace ideals in a one-dimensional Cohen-Macaulay local ring. The main theorem generalizes both of the results of S. Goto, N. Matsuoka, and T. T. Phuong ([20, Theorem 5.1]) and T. Kobayashi ([30, Theorem 1.3]) concerning the endomorphism algebra of its maximal ideal. We also explore the question of when the category of reflexive modules is of finite type, i.e., the base ring has only finitely many isomorphism classes of indecomposable reflexive modules. We show that, if the category is of finite type, the ring is analytically unramified and has only finitely many Ulrich ideals. As a consequence, Arf local rings contain only finitely many Ulrich ideals once the normalization is a local ring.

Some characterizations of relative sequentially Cohen–Macaulay and relative Cohen–Macaulay modules

Let [Formula: see text] be an [Formula: see text]-module over a Noetherian ring [Formula: see text] and [Formula: see text] an ideal of [Formula: see text] with c = [Formula: see text] ([Formula: see text], R). First, over an [Formula: see text]-relative Cohen–Macaulay local ring [Formula: see text], we provide a characterization of the [Formula: see text]-relative sequentially Cohen–Macaulay modules [Formula: see text] in terms of [Formula: see text]-relative Cohen–Macaulayness of the [Formula: see text]-modules [Formula: see text] for all [Formula: see text], where [Formula: see text]. Next, we prove that [Formula: see text] is finite [Formula: see text]-relative Cohen–Macaulay if and only if [Formula: see text] for all [Formula: see text] and [Formula: see text]. Finally, we provide another characterization of [Formula: see text]-relative sequentially Cohen–Macaulay modules [Formula: see text] in terms of vanishing of the local homology modules [Formula: see text] for all [Formula: see text] and for all [Formula: see text].

On Auslander's depth formula

We show that if Auslander's depth formula holds for non-zero Tor-independent modules over Cohen-Macaulay local rings of dimension 1, then it holds for such modules over any Cohen-Macaulay local ring. More generally, we show that the depth formula for non-zero Tor-independent modules that have finite Cohen-Macaulay dimension over depth 1 local rings implies the depth formula for such modules over any positive depth local ring.

Ratliff-Rush filtration, Hilbert coefficients and reduction number of integrally closed ideals

Let (R,m) be a Cohen-Macaulay local ring of dimension d≥3 and I an integrally closed m-primary ideal. We establish bounds for the third Hilbert coefficient e3(I) in terms of the lower Hilbert coefficients ei(I),0≤i≤2 and the reduction number of I. When d=3, the boundary cases of these bounds characterize certain properties of the Ratliff-Rush filtration of I. These properties, though weaker than depthG(I)≥1, guarantee that Rossi's bound for reduction number rJ(I) holds in dimension three. In that context, we prove that if depthG(I)≥d−3, then rJ(I)≤e1(I)−e0(I)+ℓ(R/I)+1+e2(I)(e2(I)−e1(I)+e0(I)−ℓ(R/I))−e3(I). We also discuss the signature of the fourth Hilbert coefficient e4(I).

ON THE SECOND HILBERT COEFFICIENTS AND COHEN-MACAULAY RINGS

We explore the behavior of the second Hilbert coefficients of parameter ideals in Noetherian local rings. In this paper, we provide characterizations of a Cohen-Macaulay local ring in terms of the second Hilbert coefficients and the index of reducibility of parameter ideals.

Bounds for the reduction number of primary ideal in dimension three

Let ( R , m ) (R,\mathfrak {m}) be a Cohen-Macaulay local ring of dimension d ≥ 3 d\geq 3 and I I an m \mathfrak {m} -primary ideal of R R . Let r J ( I ) r_J(I) be the reduction number of I I with respect to a minimal reduction J J of I I . Suppose depth ⁡ G ( I ) ≥ d − 3 \operatorname {depth} G(I)\geq d-3 . We prove that r J ( I ) ≤ e 1 ( I ) − e 0 ( I ) + λ ( R / I ) + 1 + ( e 2 ( I ) − 1 ) e 2 ( I ) − e 3 ( I ) r_J(I)\leq e_1(I)-e_0(I)+\lambda (R/I)+1+(e_2(I)-1)e_2(I)-e_3(I) , where e i ( I ) e_i(I) are Hilbert coefficients. Suppose d = 3 d=3 and depth ⁡ G ( I t ) > 0 \operatorname {depth} G(I^t)>0 for some t ≥ 1 t\geq 1 . Then we prove that r J ( I ) ≤ e 1 ( I ) − e 0 ( I ) + λ ( R / I ) + t r_J(I)\leq e_1(I)-e_0(I)+\lambda (R/I)+t .

On coefficient ideals

Let be a Cohen-Macaulay local ring of dimension with infinite residue field and let I be an primary ideal. Let For let be the ith-coefficient ideal of I. Also let denote the Ratliff-Rush closure of A. Let be the associated graded ring of I. We show that if for then for all . In particular if G is generalized Cohen-Macaulay then for all . As a consequence we get that if A is an analytically unramified domain with G generalized Cohen-Macaulay, then the -ification of the Rees algebra A[It] is .

Complexity and rigidity of Ulrich modules, and some applications

We analyze whether Ulrich modules, not necessarily maximal CM (Cohen-Macaulay), can be used as test modules, which detect finite homological dimensions of modules. We prove that Ulrich modules over CM local rings have maximal complexity and curvature. Various new characterizations of local rings are provided in terms of Ulrich modules. We show that every Ulrich module of dimension $s$ over a local ring is $(s+1)$-Tor-rigid-test, but not $s$−Tor-rigid in general (where $s\ge 1$). Over a deformation of a CM local ring of minimal multiplicity, we also study Tor rigidity.

Integrally Closed Ideals of Reduction Number Three

In a Cohen-Macaulay local ring [Formula: see text], we study the Hilbert function of an integrally closed [Formula: see text]-primary ideal [Formula: see text] whose reduction number is three. With a mild assumption we give an inequalityi [Formula: see text], where [Formula: see text] denotes the [Formula: see text]th Hilbert coefficient and [Formula: see text] denotes a minimal reduction of [Formula: see text]. The inequality is located between inequalities of Itoh and Elias-Valla. Furthermore, this inequality becomes an equality if and only if the depth of the associated graded ring of [Formula: see text] is larger than or equal to [Formula: see text]. We also study the Cohen-Macaulayness of the associated graded rings of determinantal rings.

On the Frobenius closure of parameter ideals in local rings F-injective on the punctured spectrum

Let be an excellent generalized Cohen-Macaulay local ring of dimension d that is F-injective on the punctured spectrum. Let be a standard parameter ideal of R. The aim of the paper is to prove that Moreover, if is contained in a large enough power of , we have

Bounds on the minimal number of generators of the dual module

Let [Formula: see text] be a Cohen–Macaulay local ring. Let [Formula: see text] be a finitely generated [Formula: see text]-module and let [Formula: see text] denote the [Formula: see text]-dual of [Formula: see text]. Furthermore, if [Formula: see text] is a maximal Cohen–Macaulay [Formula: see text]-module, then we prove that [Formula: see text], where [Formula: see text] is the cardinality of a minimal generating set of [Formula: see text] as an [Formula: see text]-module and [Formula: see text] is the multiplicity of the local ring [Formula: see text]. Furthermore, if [Formula: see text] is a reflexive [Formula: see text]-module then [Formula: see text]. As an application, we study the bound on the minimal number of generators of specific modules over two-dimensional normal local rings. We also mention some relevant examples.

Stable trace ideals and applications

We study stable trace ideals in one dimensional local Cohen–Macaulay rings and give numerous applications.

On Hilbert coefficients and sequentially generalized Cohen–Macaulay modules

This paper shows that if [Formula: see text] is a homomorphic image of a Cohen–Macaulay local ring, then [Formula: see text]-module [Formula: see text] is sequentially generalized Cohen–Macaulay if and only if the difference between Hilbert coefficients and arithmetic degrees for all distinguished parameter ideals of [Formula: see text] are bounded.