SOME CHARACTERIZATIONS OF COHEN-MACAULAY MODULES IN DIMENSION > s

On the Dimension of Formal Fibres of a Local Ring

Let A be a noetherian local ring with dimension d and I be an ideal of A . Let E=( E n )n≥ 0 be a good I-filtration of submodules of an A-module E . Let H be an ideal of A containing I and F H E) = ⨁n ≥ 0 E n / HE n . Assume that E is a Cohen–Macaulay module with λ(E/IE) finite and Ann(E) = 0, and let J be a minimal reduction of I. In this paper we give conditions on λ(E n ∩ JE/JE n−1) and λ(HE n ∩ JE/JHE n−1) so, that F H (E), has depth of at least d − 1.

Let (R,𝔪) be a Noetherian local ring, M a finitely generated R-module with finite projective dimension n, N an arbitrary R-module, and 𝔞 be an ideal of R which is generated by s elements. In this article, we provide a surjective homomorphism from ordinary local cohomology module to top generalized local cohomology module , where P n, M is an nth syzygy of a projective resolution of M. Also, by using this epimorphism, we prove some results about the attached primes, coassociated primes, the Betti numbers, and Artinian properties of certain generalized local cohomology modules.

Let (R, 𝔪) be a Noetherian local ring and M a finitely generated R-module with dim M = d. This paper is concerned with the following property for the top local cohomology module : In this article we will show that this property is equivalent to the catenaricity of the unmixed support USupp M of M which is defined by Usupp M = Supp M/UM(0), where UM(0) is the largest submodule of M of dimension less than d. Some characterizations of this property in terms of system of parameters as well as the relation between the unmixed supports of M and of the 𝔪-adic completion are given.

Let R be a commutative ring with identity. A filtration on R is a decreasing sequence {In}∞n=0 of ideals of R. Associated to a filtration is a well-defined completion R∗ = lim ←− R/In and a canonical homomorphism ψ : R → R ∗ [13, Chap. 9]. If ⋂∞ n=0 In = (0), then ψ is injective and R may be regarded as a subring of R∗ [13, p. 401]. In the terminology of Northcott, a filtration {In}∞n=0 is multiplicative if I0 = R and InIm ⊆ In+m for all m ≥ 0 and n ≥ 0 [13, p. 408]. A well-known example of a multiplicative filtration on R is the I -adic filtration {I n}∞n=0, where I is a fixed ideal of R. In this paper we consider filtrations of ideals of R that are not multiplicative and examine the completions associated to these filtrations. We assume the ring R is Noetherian. Instead of successive powers of a fixed ideal I, we use a filtration formed from a more general descending sequence {In}∞n=0 of ideals. We require, for each n > 0, that the nth ideal In be contained in the nth power of the Jacobson radical of R and that Ink ⊆ I k n for all k, n ≥ 0. We call the associated completion a multi-adic completion and denote it by R∗. The basics of the multi-adic construction and the relationship between this completion and certain ideal-adic completions are considered in Section 2. In Section 3 we prove, for {In} as just described, that R∗ is Noetherian. Let (R, m) be a Noetherian local ring. If R is excellent, Henselian, or universally catenary, we prove in Section 4 that R∗ has the same property. We were inspired to pursue this project partly because of our continuing interest in exploring completions and power series. In our previous work we constructed various examples of rings inside relatively well-understood rings such as the (x)adic completion k[y][[x]] of a polynomial ring k[x, y] in two variables x and y over a field k [4; 5]. The examples we obtained demonstrate that certain properties of a ring may fail to extend to its m-adic completion, where m is a maximal ideal [6]. The process of passing to completion gives an analytic flavor to algebra. Often we view completions in terms of power series or in terms of coherent sequences, as

Suppose (R, m) is a local Noetherian domain with quotient field K and m-adic completion Ȓ. It is well known that the fibers of the morphism Spec(Ȓ) ₒ Spec(R), i.e., the formal fibers of R, encode important information about the structure of R. Perhaps the most important condition in Grothendieck’s definition of R being excellent is that the formal fibers of R be geometrically regular. Indeed, a local Noetherian ring is excellent provided it is universally catenary and has geometrically regular formal fibers [G, (7.8.3), page 214]. But the structure of the formal fibers of R is often difficult to determine. We are interested here in bringing out the interrelatedness of properties of the generic formal fiber of R with the existence of certain local Noetherian domains C birationally dominating R and having C/mC is a finite R-module.

Local cohomology modules via certain flat extension rings

On Buchsbaum rings

A commutative Noetherian local ring of finite global dimension is a regular local ring and the structure of such a ring is well described 1n many texts (see §1.9). It is perhaps surprising that very little seems to be known about the corresponding non-commutat1ve rings and it is our aim 1n this thesis to examine some of the ways in which the commutative theory can, and cannot be extended to a right Noetherian local ring of finite right global dimension. Chapter 1 contains basic definitions and, in Chapter 2, results are obtained on the projective and Injective dimensions of modules over right Noetherian local rings. Commutative regular local rings are domains and we begin Chapter 3 by considering the question of when a right Noetherian local ring R of finite right global dimension is a prime ring. An example of Stafford's shows that R is not necessarily prime; however, by examining the Nilpotent radical of R, we are able to extend results of Ramras [57] and Walker [75] and show that R is indeed prime when certain prime ideals are localisable. In Chapter 4 we consider the lattice of prime ideals of R when R is an AR-r1ng and provide theorems which generalise some of the basic results from the theory of commutative regular local rings. Section 4.3 contains examples which not only illustrate points arising in Chapters 3 and 4 but also show that some of the techniques which are mainstays of the commutative theory, fall dramatically in a non-commutat1ve setting. In Chapter 5, we generalise the concepts of regular sequences and Cohen Macaulay rings enabling us to prove, in §5.2 and chapter 6, that a right Noetherian local ring of finite right global dimension, which 1s integral over its centre, is a prime ring and exhibits many properties similar to those enjoyed by commutative regular local rings. Examples are provided which show that some alternative approaches are not applicable to the rings considered in these chapters. Regular local rings are Gorensteln rings and, as such, are the subject of an elegant structure theorem due to Bass [5]. In Chapter 7, we generalise his theorem to the situation of rings Integral over their centres. Each chapter begins with a summary of the results contained in that chapter.

Let [Formula: see text] be a Noetherian local ring, [Formula: see text] two ideals of [Formula: see text], and [Formula: see text] two finitely generated [Formula: see text]-modules. It is first shown that [Formula: see text] is a finite set. We also prove that except the maximal ideal [Formula: see text], the set [Formula: see text] is stable for large [Formula: see text], where we use [Formula: see text] to denote [Formula: see text]-module [Formula: see text] or [Formula: see text] and [Formula: see text] is the eventual value of [Formula: see text].

Given a Noetherian local ring ( R , m ) (R,m) it is shown that there exists an integer ℓ \ell such that R R is Gorenstein if and only if some system of parameters contained in m ℓ m^{\ell } generates an irreducible ideal. We obtain as a corollary that R R is Gorenstein if and only if every power of the maximal ideal contains an irreducible parameter ideal.

Let $$(R,\mathfrak {m})$$ be a commutative Noetherian local ring, M be a finitely generated R-module and $$\mathfrak {a}$$ , I and J are ideals of R. We investigate the structure of formal local cohomology modules of $$\mathfrak {F}^i_{\mathfrak {a},I,J}(M)$$ and $$\check{\mathfrak {F}}^i_{\mathfrak {a},I,J}(M)$$ with respect to a pair of ideals, for all $$i\ge 0$$ . The main subject of the paper is to study the finiteness properties and artinianness of $$\mathfrak {F}^i_{\mathfrak {a},I,J}(M)$$ and $$\check{\mathfrak {F}}^i_{\mathfrak {a},\mathfrak {m},J}(M)$$ . We study the maximum and minimum integer $$i\in \mathbb {N}$$ such that $$\mathfrak {F}^i_{\mathfrak {a},\mathfrak {m},J}(M)$$ and $$\check{\mathfrak {F}}^i_{\mathfrak {a},\mathfrak {m},J}(M)$$ are not Artinian and we obtain some results involving cosupport, coassociated and attached primes for formal local cohomology modules with respect to a pair of ideals. Also, we give an criterion involving the concepts of finiteness and vanishing of formal local cohomology modules and Cech-formal local cohomology modules with respect to a pair of ideals.

Let $$(R,\mathfrak {m})$$ be a Noetherian local ring of prime characteristic p and Q be an $$\mathfrak {m}$$ -primary parameter ideal. We give criteria for F-rationality of R using the tight Hilbert function $$H^*_Q(n)=\ell (R/(Q^n)^*)$$ and the coefficient $$e_1^*(Q)$$ of the tight Hilbert polynomial $$P^*_Q(n)=\sum _{i=0}^d(-1)^ie_i^*(Q)\left( {\begin{array}{c}n+d-1-i\\ d-i\end{array}}\right) .$$ We obtain a lower bound for the tight Hilbert function of Q for equidimensional excellent local rings that generalizes a result of Goto and Nakamura. We show that if $$\dim R=2 $$ , the Hochster–Huneke graph of R is connected and this lower bound is achieved, then R is F-rational. Craig Huneke asked if the F-rationality of unmixed local rings may be characterized by the vanishing of $$e_1^*(Q).$$ We construct examples to show that without additional conditions, this is not possible. Let R be an excellent, reduced, equidimensional Noetherian local ring and Q be generated by parameter test elements. We find formulas for $$e_1^*(Q), e_2^*(Q), \ldots , e_d^*(Q)$$ in terms of Hilbert coefficients of Q, lengths of local cohomology modules of R, and the length of the tight closure of the zero submodule of $$H^d_{\mathfrak {m}}(R).$$ Using these, we prove: R is F-rational $$\iff e_1^*(Q)=e_1(Q) \iff {\text {depth}}R\ge 2$$ and $$e_1^*(Q)=0.$$

Let be a Noetherian local ring, I be an ideal of R and M be a finitely generated R-module such that is Artinian and I-cofinite, where In this paper, we establish shifted principles under localization and completion for attached primes of the top local cohomology module

Local cohomology based on a nonclosed support defined by a pair of ideals