Local Homology Modules Research Articles

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].

Coartinianess of local homology modules for ideals of small dimension

Let [Formula: see text] be an ideal of a commutative noetherian ring [Formula: see text] and [Formula: see text] an [Formula: see text]-module with Cosupport in [Formula: see text]. We show that [Formula: see text] is [Formula: see text]-coartinian if and only if [Formula: see text] is artinian for all [Formula: see text], which provides finite steps to examine [Formula: see text]-coartinianess. We also consider the duality of Hartshorne’s questions: for which rings [Formula: see text] and ideals [Formula: see text] are the modules [Formula: see text] [Formula: see text]-coartinian for every artinian [Formula: see text]-module [Formula: see text] and all [Formula: see text]; whether the category [Formula: see text] of [Formula: see text]-coartinian modules is an abelian subcategory of the category of [Formula: see text]-modules, and establish affirmative answers to these questions in the case [Formula: see text] and [Formula: see text].

CFA modules and the finiteness of coassociated primes of local homology modules

We introduce the concept CFA modules and their applications in investigation the coassociated primes of local homology modules. The main result of this paper says that if $M$ is a CFA linearly compact $R$-module and $t$ is a non-negative integer such that H i I ( M ) is CFA for all $i < t$, then R / I ⊗ R H t I ( M ) is CFA. Hence, the set $\mathrm{Coass}_R$ H t I ( M ) is finite.

The nice m-system of parameters for Artinian modules

This paper restates the definition of the nice m-system of parameters for Artinian modules. It also shows its effects on the differences between lengths and multiplicities of certain systems of parameters for Artinian modules: In particular, if is a nice m-system of parameters then the function is a polynomial having very nice form. Moreover, we will prove some properties of the nice m-system of parameters for Artinian modules. Especially, its effect on the annihilation of local homology modules of Artinian module A.

Local homology, Koszul homology and Serre classes

Given a Serre class $\mathcal {S}$ of modules, we compare the containment of Koszul homology, Ext modules, Tor modules, local homology and local cohomology in $\mathcal {S}$ up to a given bound $s \geq 0$. As applications, we give a full characterization of Noetherian local homology modules. Furthermore, we establish a comprehensive vanishing result which readily leads to formerly known descriptions of the numerical invariants' width and depth in terms of Koszul homology, local homology and local cohomology. In addition, we recover a few renowned vanishing criteria scattered throughout the literature.

DUAL OF FALTINGS’ THEOREMS ON FINITENESS OF LOCAL COHOMOLOGY

Let $R$ be a commutative Noetherian ring and $\fa$ an ideal of $R$. We intend to establish the dual of two Faltings' Theorems for local homology modules of an Artinian module. As a consequence of this, we show that, if $A$ is an Artinian module over semi-local complete ring $R$ and $j$ is an integer such that $H_i^{\fa}(A)$ is Artinian for all $i<j$, then the set $\Coass_R(H_j^{\fa}(A))$ is finite. We also introduce the notion of the $n$th Artinianness dimension $g_n^\fa(A)=\inf\{g^{\fa R_{\fp}}(^\fp A): \fp\in\Spec(R) \text{and} \dim R/\fp\geq n\}$, for all $n\in\mathbb{N}_{0}$ and prove that $g_1^\fa(A)=\inf\{i\in\mathbb{N}_0: H_i^\fa(A) \text{is not minimax}\}$, whenever $R$ is a semi-local complete ring. Moreover, in this situation we show that $\Coass_R(H_{g_1^\fa(A)}^\fa(A))$ is a finite set.

Minimaxness and finiteness properties of local homology and local cohomology modules

We prove some results concerning minimaxness and finiteness of local homology modules and by Matlis duality we extend some results for the minimaxness and finiteness of local cohomology modules. We introduce the concept of C-minimax R-modules, and we discuss the maximum and minimum integers such that local homology and local cohomology modules are C-minimax. As a consequence, we find minimum integers such that local homology and local cohomology modules are of finite length.

Generalized Local Homology Modules of Complexes

Generalized Local Homology Modules of Complexes

Extended Local Cohomology and Local Homology

We present an in-depth exploration of the module structures of local (co)homology modules (moreover, for complexes) over the completion \(\widehat {R}^{\mathcal {a}}\) of a commutative noetherian ring R with respect to a proper ideal \(\mathcal {a}\). In particular, we extend Greenlees-May Duality and MGM Equivalence to track behavior over \(\widehat {R}^{\mathcal {a}}\), not just over R. We apply this to the study of two recent versions of homological finiteness for complexes, and to certain isomorphisms, with a view toward further applications. We also discuss subtleties and simplifications in the computations of these functors.

Minimax modules, local homology and local cohomology

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 [Formula: see text]-separated modules and of minimax local homology modules. By duality, we get some properties of Grothendieck’s local cohomology modules.

The finiteness of coassociated primes of generalized local homology modules

We present some finiteness results for co-associated primes of generalized local homology modules. Let M be a finitely generated R-module and N a linearly compact R-module. If N and HiI (N) satisfy the finiteness condition for co-associated primes for all i < k, then CoassR(HkI (M,N)) is a finite set. On the other hand, if HiI (N) = 0 for all i < t and TorjR (M,Hti (N)) = 0 for all j < h, then TorhR (M,HtI (N)) ≅ Hh+tI(M,N). Moreover, Coass(Hh+tI(M,N)) is also a finite set providedN satisfies the finiteness condition for co-associated primes. Finally, N is a semi-discrete linearly compact R-module such that 0:NI ≠ 0. Let t=WidthI (N) and h = tor_(M,HtI (N)); it follows that WidthI+Ann(M)(N) = t + h and Coass(Hh+tI(M,N)) is a finite set.

SOME PROPERTIES OF GENERALIZED LOCAL HOMOLOGY AND COHOMOLOGY MODULES

We study some properties of representable generalized local homology modules. By duality, we get some properties of good generalized local cohomology modules.

Some properties of local homology and local cohomology modules

We study some properties of representable or I-stable local homology modules HiI (M) where M is a linearly compact module. By duality, we get some properties of good or at local cohomology modules HIi (M) of A. Grothendieck.

ON THE VANISHING OF LOCAL HOMOLOGY MODULES

AbstractLet R be a commutative Noetherian ring, is an ideal of R and M is an R-module. We intend to establish the dual of Grothendieck's Vanishing Theorem for local homology modules. We conjecture that =0 for all i&gt;magRM. We prove this in several cases.

Some results on local homology and local cohomology modules

In this paper, we obtain some results about the local homology modules of Artinian modules, and by Matlis duality we obtain some results about the local cohomology modules of finitely generated modules.

Generalized Local Homology for Artinian Modules

We introduce generalized local homology which is in some sense dual to generalized local cohomology, and study some properties of generalized local homology modules for artinian modules, such as the artinianness, noetherianness and the characterization of Width I(M) by generalized local homology. By using duality, we get back some properties of generalized local cohomology.

Coassociated primes of local homology and local cohomology modules

Coassociated primes of local homology and local cohomology modules

A Finiteness Result for Co-Associated and Associated Primes of Generalized Local Homology and Cohomology Modules

We show that if M is a finitely generated R-module, N is an I-stable semi-discrete linearly compact R-module and G is a closed R-submodule of the generalized local homology module such that is I-stable for all j < i and is I-stable, then the set Coass G is finite. As an immediate consequence, the first non-I-stable generalized local homology module of N, M with respect to I has only finitely many co-associated primes. By duality, we get a new result for the finiteness of associated primes of (generalized) local cohomology modules.

A local homology theory for linearly compact modules

We introduce a local homology theory for linearly compact modules which is in some sense dual to the local cohomology theory of A. Grothendieck [A. Grothendieck, Local Cohomology, Lecture Notes in Math., vol. 20, Springer-Verlag, Berlin/Tokyo/New York, 1967. [10]]. Some basic properties such as the noetherianness, the vanishing and non-vanishing of local homology modules of linearly compact modules are proved. A duality theory between local homology and local cohomology modules of linearly compact modules is developed by using Matlis duality and Macdonald duality. As consequences of the duality theorem we obtain some generalizations of well-known results in the theory of local cohomology for semi-discrete linearly compact modules.

On the finiteness of co-associated primes of local homology modules

We show that if a lineary compact $R$-module $M$ and local homology modules $H^I_i(M)$ satisfy the finite condition for co-associated primes for all $i<s$, then the set $\textrm{Coass}_R (H^I_s(M))$ is finite.