Narrow Results

Let $R$ be a polynomial ring in $n$ indeterminates with coefficients in a field $K$ of characteristic $p>0,$ and ${𝒟}$ be the ring of differential operators over $R.$ We consider $𝔭$ as a nonmaximal homogeneous prime ideal, and ${}^{\ast }E(R/𝔭)$ as a graded injective hull of $R/𝔭.$ In this paper, we prove the generalized Eulerian property for the graded injective hull $R/𝔭$ as a ${𝒟}$-module in positive characteristic. Furthermore, we show that the minimal injective resolution of $\mathrm{\text{F}}$-finite module ends in an object which is the direct sum of indecomposable in graded case using some finiteness properties in prime characteristic.

In [D. Rees, Hilbert functions and pseudo-rational local rings of dimension two, J. London Math. Soc. (2) 24 (1981) 467–479], Rees gave a characterization for the normal joint reduction number zero of two $𝔪$-primary ideals in an analytically unramified Cohen–Macaulay local ring of dimension two. Rees’ result is a generalization of Zariski’s product theorem for complete ideals in a regular local ring of dimension two. The aim of this paper is to extend Rees’ theorem for the ordinary powers of $𝔪$-primary ideals $I$ and $J$ in a Cohen–Macaulay local ring of dimension two. Following Rees’ approach, we define the modified Koszul homology modules $M_{r,s}^1(a^k,b^k)$ for a joint reduction $(a,b)$ of $I$ and $J$. Under the additional assumption that the associated graded rings of $I$ and $J$ have positive depth, we obtain a characterization of the joint reduction number zero of $I$ and $J$ in terms of the vanishing of the module $M_{0,0}^1(a,b)$, as well as in terms of the Hilbert coefficients and the bigraded Hilbert coefficients. More generally, we introduce the joint reduction lattice and study the vanishing of $M_{r,s}^1(a,b)$ for any $r,s\ge 0$. This gives a characterization for a vector $(r,s)$ to be in the joint reduction lattice of $I$ and $J$. We also give a cohomological interpretation of these theorems by investigating the local cohomology modules of the bigraded extended Rees algebra. This gives another characterization for a vector $(r,s)$ to be in the joint reduction lattice and also extends a recent result of Masuti and Verma in [Local cohomology of bigraded Rees algebras and normal Hilbert coefficients, J. Pure Appl. Algebra218(5) (2014) 904–918] for ordinary powers of ideals.

We show some results about local homology modules and local cohomology modules concerning Grothendieck’s conjecture and Huneke’s question. We also show some equivalent properties of $I$-separated modules and of minimax local homology modules. By duality, we get some properties of Grothendieck’s local cohomology modules.

We study the generalized local homology for linearly compact modules which is a generalization of the local homology theory. By duality, we get some properties of the generalized local cohomology and extend well-known properties of the local cohomology theory of Grothendieck.

The goal of this paper is twofold: on the one hand, motivated by questions raised by Schenzel, we explore situations where the Hartshorne–Lichtenbaum Vanishing theorem for local cohomology fails, leading us to simpler expressions of certain local cohomology modules. As application, we give new expressions of the endomorphism ring of these modules. On the other hand, building upon previous work by Àlvarez Montaner, we exhibit the shape of Lyubeznik tables of the so-called partially sequentially Cohen–Macaulay rings as introduced by Sbarra and Strazzanti.

The aim of this paper is to introduce a new notion of sequences called dd-sequences and show that this notion may be convenient for studying the polynomial property of partial Euler–Poincaré characteristics of the Koszul complex with respect to the powers of a system of parameters. Some results about the dd-sequences, the partial Euler–Poincaré characteristics and the lengths of local cohomology modules are also presented in the paper. There are also applications of dd-sequences on the structure of sequentially Cohen–Macaulay modules.

This paper is concerned with the relationships between two concepts, vanishing of cohomology groups and the structure of free resolutions. In particular, we study the connection between vanishing theorems for the local cohomology of multigraded modules and the structure of their free multigraded resolutions.

Let I be an ideal of a commutative Noetherian ring R such that ara(I) = t ≥ 2. The purpose of this article is to show that there exists an I-filter regular sequence y₁, …, y_t for R such that Rad(I) = Rad(y₁, …, y_t) and cd((y₁, …, y_i), R) = i for all 1 ≤ i < t. Also, it is shown that ara(I) ≤ dim R + 1, which is a generalization of a nice result of Kronecker [14]. In addition, some applications are included.

Let f₀, f₁, f₂, f₃ be linearly independent nonzero homogeneous polynomials in the standard ℤ-graded ring R ≔ 𝕂[s, t, u] of the same degree d, and gcd(f₀, f₁, f₂, f₃) = 1. This defines a rational map ℙ² → ℙ³. The Rees algebra Rees(I) = R ⊕ I ⊕ I² ⊕ ⋯ of the ideal I = 〈f₀, f₁, f₂, f₃〉 is the graded R-algebra which can be described as the image of the R-algebra homomorphism h: R[x, y, z, w ] → Rees(I). This paper discusses one result concerning the structure of the kernel of the map h when I is a saturated local complete intersection ideal with V(I) ≠ ∅ and μ-basis of degrees (1,1,d - 2).

Let R be a commutative Noetherian ring, 𝔞 be an ideal of R and M be an arbitrary R-module. In this paper, among other things, we show that if, for a non-negative integer t, the R-module is weakly Laskerian and is 𝔞-weakly cofinite for all i < t, then, for any weakly Laskerian submodule U of , the R-module is weakly Laskerian. As a consequence the set of associated primes of is finite.

Let R be a commutative regular ring of dimension d, I be an ideal of R and M be a non-zero finitely generated R-module. In this paper we show that Huneke's two different conjectures are equivalent. Also we provide some partially affirmative answers to them. In fact it is shown that the Bass numbers of are finite for all i ≥ 0, whenever d ≤ 3. Also if (R, m) be regular local ring, we show that the Bass numbers are finite, for all i ≥ 0 and all j ≥ 0, and , for all i ≥ 0 and all j ≥ 0, whenever height(I) = 1.

Let R be a commutative Noetherian local ring and let 𝔞 be a proper ideal of R. In this paper, as a main result, it is shown that if M is a Gorenstein R-module with c = ht_M𝔞, then for all i ≠ c is completely encoded in homological properties of , in particular in its Bass numbers. Notice that, this result provides a generalization of a result of Hellus and Schenzel which has been proved before, as a main result, in the case where M = R.

Let (R, 𝔪, k) be a complete Gorenstein local ring of dimension n. Let be the local cohomology module with respect to a pair of ideals I, J and . In this paper we will show that the endomorphism ring is a commutative ring. In particular if for all i ≠ t, then B is isomorphic to R. Also we prove that, B is a finite R-module if and only if is an Artinian R-module, where d = n - t. Moreover we will show that in the case that for all i ≠ t the natural homomorphism is nonzero which gives a positive answer to a conjecture due to Hellus–Schenzel (see [On cohomologically complete intersections, J. Algebra 320 (2008) 3733–3748]).

Let $R$ be a commutative Noetherian ring, $d\ge 0$ be an integer and let $\sum$ denote the ideals of $R$ of dimension $\le d$. For an integer $i\ge 0$ and $R$-modules $M,N$, we define the $i$th cohomology module $H_d^i(M,N)={\underrightarrow{lim}}_{𝔞\in \sum }{\mathrm{\text{Ext}}}_R^i(M/𝔞M,N)$ and study some of its properties. Some vanishing results will be proved and the connection between the vanishing of this cohomology modules with the dimension of certain subsets of $\mathrm{\text{Supp}}(M)$ will be investigated.

In this paper we present a technical lemma about localization at countably infinitely many prime ideals. We apply this lemma to get many results about the finiteness of associated prime ideals of local cohomology modules.

Let $(R,𝔪)$ be a commutative Noetherian local ring, which is a homomorphic image of a Gorenstein local ring and $I$ an ideal of $R$. Let $M$ be a nonzero finitely generated $R$-module and $i\ge 0$ be an integer. In this paper we show that, the $R$-module $H_{𝔪}^i(M)$ is nonzero and $I$-cofinite if and only if $Rad(I+0{:}_R{H}_{𝔪}^i(M))=𝔪$. Also, several applications of this result will be included.

Let $𝔞$ be an ideal of a commutative noetherian ring $R$ with unity and $M$ an $R$-module supported at $\mathrm{\text{V}}(𝔞)$. Let $n$ be the supermum of the integers $i$ for which $H_i^{𝔞}(M)\ne 0$. We show that $M$ is $𝔞$-cofinite if and only if the $R$-module ${Tor}_i^R(R/𝔞,M)$ is finitely generated for every $0\le i\le n$. This provides a hands-on and computable finitely-many-steps criterion to examine $𝔞$-confiniteness. Our approach relies heavily on the theory of local homology which demonstrates the effectiveness and indispensability of this tool.

Let $R$ be a commutative Noetherian ring, $I$ and $J$ be two ideals of $R$ and $M$ be an $R$-module (not necessary $I$-torsion). In this paper among other things, it is shown that if dim $M\le 1$, then the $R$-module ${\mathrm{\text{Ext}}}_R^i(R/I,M)$ is finitely generated, for all $i\ge 0$, if and only if the $R$-module ${\mathrm{\text{Ext}}}_R^i(R/I,M)$ is finitely generated, for $i=0,1$. As a consequence, we prove that if $M$ is finitely generated and $t\in \mathbb{N}$ such that the $R$-module ${\mathrm{\text{H}}}_{I,J}^i(M)$ is ${\mathrm{\text{FD}}}_{\le 1}$ (or weakly Laskerian) for all $i<t$, then ${\mathrm{\text{H}}}_{I,J}^i(M)$ is $(I,J)$-cofinite for all $i<t$ and for any ${\mathrm{\text{FD}}}_{\le 0}$ (or minimax) submodule $N$ of ${\mathrm{\text{H}}}_{I,J}^t(M)$, the $R$-modules ${\mathrm{\text{Hom}}}_R(R/I,{\mathrm{\text{H}}}_{I,J}^t(M)/N)$ and ${\mathrm{\text{Ext}}}_R^1(R/I,{\mathrm{\text{H}}}_{I,J}^t(M)/N)$ are finitely generated. Also it is shown that if dim $M/𝔞M\le 1$ (e.g. dim $R/𝔞\le 1$) for all $𝔞\in \tilde{W}(I,J)$, then the local cohomology module ${\mathrm{\text{H}}}_{I,J}^i(M)$ is $(I,J)$-cofinite for all $i\ge 0$.

The purpose of the present paper is to continue the study of modules cofinite and weakly cofinite with respect to an ideal $𝔞$ of a Noetherian ring $R$. It is shown that an $R$-module $M$ is cofinite with respect to $𝔞$, if and only if, ${Ext}_R^i(R/𝔞,M)$ is finitely generated for all $i\le \mathrm{\text{cd}}(𝔞,M)+1$, whenever $dimR/𝔞=1$. In addition, we show that if $M$ is finitely generated and $H_{𝔞}^i(M)$ are weakly Laskerian for all $i\le t-1$, then $H_{𝔞}^i(M)$ are $𝔞$-cofinite for all $i\le t-1$ and for any minimax submodule $K$ of $H_{𝔞}^t(M)$, the $R$-modules ${Hom}_R(R/𝔞,H_{𝔞}^t(M)/K)$ and ${Ext}_R^1(R/𝔞,H_{𝔞}^t(M)/K)$ are finitely generated, where $t$ is a non-negative integer. Finally, we explore a criterion for weakly cofiniteness of modules with respect to an ideal of dimension one. Namely, for such ideals it suffices that the two first $Ext$-modules in the definition for weakly cofiniteness are weakly Laskerian. As an application of this result, we deduce that the category of all $𝔞$-weakly cofinite modules over $R$ forms a full Abelian subcategory of the category of modules.

In this paper, we introduce a generalization of the ordinary ideal transform, denoted by $D_{I,J}$, which is called the ideal transform with respect to a pair of ideals $(I,J)$ and has an apparent algebraic structure. Then we study its various properties and explore the connection with the ordinary ideal transform. Also, we discuss the associated primes of local cohomology modules with respect to a pair of ideals. In particular, we give a characterization for the associated primes of the nonvanishing generalized local cohomology modules.