scholarly journals On the dimension and multiplicity of local cohomology modules

2002 ◽  
Vol 167 ◽  
pp. 217-233 ◽  
Author(s):  
Markus P. Brodmann ◽  
Rodney Y. Sharp
Keyword(s):  
Local Ring ◽  
Maximal Ideal ◽  
Local Cohomology ◽  
Local Rings ◽  
Zariski Topology ◽  
Finitely Generated Module ◽  
Local Cohomology Modules ◽  
Matlis Duality ◽  
Gorenstein Local Ring ◽  
Cohomology Modules

AbstractThis paper is concerned with a finitely generated module M over a (commutative Noetherian) local ring R. In the case when R is a homomorphic image of a Gorenstein local ring, one can use the well-known associativity formula for multiplicities, together with local duality and Matlis duality, to produce analogous associativity formulae for the local cohomology modules of M with respect to the maximal ideal. The main purpose of this paper is to show that these formulae also hold in the case when R is universally catenary and such that all its formal fibres are Cohen–Macaulay.These formulae involve certain subsets of the spectrum of R called the pseudosupports of M; these pseudo-supports are closed in the Zariski topology when R is universally catenary and has the property that all its formal fibres are Cohen–Macaulay. However, examples are provided to show that, in general, these pseudo-supports need not be closed. We are able to conclude that the above-mentioned associativity formulae for local cohomology modules do not hold over all local rings.

On the cofiniteness of Artinian local cohomology modules

2016 ◽  
Vol 15 (04) ◽  
pp. 1650070 ◽  
Author(s):  
Ghader Ghasemi ◽  
Kamal Bahmanpour ◽  
Jafar A’zami
Keyword(s):  
Local Ring ◽  
Homomorphic Image ◽  
Local Cohomology ◽  
Finitely Generated ◽  
Noetherian Local Ring ◽  
Local Cohomology Modules ◽  
Gorenstein Local Ring ◽  
Cohomology Modules

Let [Formula: see text] be a commutative Noetherian local ring, which is a homomorphic image of a Gorenstein local ring and [Formula: see text] an ideal of [Formula: see text]. Let [Formula: see text] be a nonzero finitely generated [Formula: see text]-module and [Formula: see text] be an integer. In this paper we show that, the [Formula: see text]-module [Formula: see text] is nonzero and [Formula: see text]-cofinite if and only if [Formula: see text]. Also, several applications of this result will be included.

Gorenstein Injective Resolutions, Cousin Complexes and Top Local Cohomology Modules

2009 ◽  
Vol 16 (01) ◽  
pp. 95-101
Author(s):  
Kazem Khashyarmanesh
Keyword(s):  
Local Ring ◽  
Local Cohomology ◽  
Local Cohomology Module ◽  
Injective Resolution ◽  
Local Cohomology Modules ◽  
Gorenstein Local Ring ◽  
Gorenstein Injective ◽  
Cousin Complexes ◽  
Cohomology Modules ◽  
Cousin Complex

Let R be a Gorenstein local ring. We show that for a balanced big Cohen–Macaulay module M over R, the Cousin complex [Formula: see text] provides a Gorenstein injective resolution of M. Also, over a d-dimensional Gorenstein local ring R with maximal ideal 𝔪, we show that [Formula: see text], the dth local cohomology module of M with respect to 𝔪, is Gorenstein injective if (a) M is a balanced big Cohen–Macaulay R-module, or (b) M ∈ G(R), where G(R) is the Auslander's G-class of R.

Cohomological annihilators

1982 ◽  
Vol 91 (3) ◽  
pp. 345-350 ◽  
Author(s):  
Peter Schenzel
Keyword(s):  
Maximal Ideal ◽  
Commutative Algebra ◽  
Local Cohomology ◽  
Local Rings ◽  
Intersection Theorem ◽  
Prime Characteristic ◽  
Local Cohomology Modules ◽  
System Of Parameters ◽  
Cohomology Modules ◽  
Unique Maximal Ideal

The local cohomology theory introduced by Grothendieck(1) is a useful tool for attacking problems in commutative algebra and algebraic geometry. Let A denote a local ring with its unique maximal ideal m. For an ideal I ⊂ A and a finitely generated A-module M we consider the local cohomology modules HiI (M), i є ℤ, of M with respect to I, see Grothendieck(1) for the definition. In particular, the vanishing resp. non-vanishing of the local cohomology modules is of a special interest. For more subtle considerations it is necessary to study the cohomological annihilators, i.e. aiI(M): = AnnΔHiI(M), iєℤ. In the case of the maximal ideal I = m these ideals were used by Roberts (6) to prove the ‘New Intersection Theorem’ for local rings of prime characteristic. Furthermore, we used this notion (7) in order to show the amiability of local rings possessing a dualizing complex. Note that the amiability of a system of parameters is the key step for Hochster's construction of big Cohen-Macaulay modules for local rings of prime characteristic, see Hochster(3) and (4).

Cofiniteness of Local Cohomology Modules

2014 ◽  
Vol 21 (04) ◽  
pp. 605-614 ◽  
Author(s):  
Kamal Bahmanpour ◽  
Reza Naghipour ◽  
Monireh Sedghi
Keyword(s):  
Local Ring ◽  
Local Cohomology ◽  
Partial Answer ◽  
Finitely Generated ◽  
Finitely Generated Module ◽  
Noetherian Local Ring ◽  
Dimension Formula ◽  
Complete Local Ring ◽  
Local Cohomology Modules ◽  
Cohomology Modules

Let M be a non-zero finitely generated module over a commutative Noetherian local ring (R, 𝔪). In this paper we consider when the local cohomology modules are finitely generated. It is shown that if t ≥ 0 is an integer and [Formula: see text], then [Formula: see text] is not 𝔭-cofinite. Then we obtain a partial answer to a question raised by Huneke. Namely, if R is a complete local ring, then [Formula: see text] is finitely generated if and only if 0 ≤ n ∉ W, where [Formula: see text]. Also, we show that if J ⊆ I are 1-dimensional ideals of R, then [Formula: see text] is J-cominimax, and [Formula: see text] is finitely generated (resp., minimax) if and only if [Formula: see text] is finitely generated for all [Formula: see text] (resp., [Formula: see text]). Moreover, the concept of the J-cofiniteness dimension [Formula: see text] of M relative to I is introduced, and we explore an interrelation between [Formula: see text] and the filter depth of M in I. Finally, we show that if R is complete and dim M/IM ≠ 0, then [Formula: see text].

scholarly journals On Vanishing of Generalized Local Cohomology Modules

2005 ◽  
Vol 12 (02) ◽  
pp. 213-218 ◽  
Author(s):  
K. Divaani-Aazar ◽  
R. Sazeedeh ◽  
M. Tousi
Keyword(s):  
Local Ring ◽  
Local Cohomology ◽  
Finitely Generated ◽  
Local Cohomology Modules ◽  
Generalized Local Cohomology Modules ◽  
Gorenstein Local Ring ◽  
Cohomology Modules

Let [Formula: see text] denote an ideal of a d-dimensional Gorenstein local ring R, and M and N two finitely generated R-modules with pd M < ∞. It is shown that [Formula: see text] if and only if [Formula: see text] for all [Formula: see text].

scholarly journals On top local cohomology modules, Matlis duality and tensor products

2019 ◽  
Vol 18 (09) ◽  
pp. 1950179
Author(s):  
M. Y. Sadeghi ◽  
M. Eghbali ◽  
Kh. Ahmadi-Amoli
Keyword(s):  
Local Ring ◽  
Local Cohomology ◽  
Tensor Products ◽  
Cohomological Dimension ◽  
Local Cohomology Modules ◽  
Matlis Duality ◽  
Cohomology Modules

Let [Formula: see text] be an ideal of a local ring [Formula: see text] with [Formula: see text] the cohomological dimension of [Formula: see text] in [Formula: see text]. In the case that [Formula: see text], we first give a bound for [Formula: see text], where [Formula: see text] and [Formula: see text] is complete. Later, [Formula: see text], [Formula: see text] and [Formula: see text] are examined. In the case [Formula: see text], the set [Formula: see text] is considered.

Top Local Cohomology Modules

2007 ◽  
Vol 14 (02) ◽  
pp. 209-214 ◽  
Author(s):  
Mohammad T. Dibaei ◽  
Siamak Yassemi
Keyword(s):  
Finite Number ◽  
Local Ring ◽  
Local Cohomology ◽  
Finitely Generated ◽  
Finitely Generated Module ◽  
Noetherian Local Ring ◽  
Local Cohomology Modules ◽  
Cohomology Modules

For a finitely generated module M over a commutative Noetherian local ring (R,𝔪), it is shown that there exist only a finite number of non-isomorphic top local cohomology modules [Formula: see text] for all ideals 𝔞 of R. It is also shown that for a given integer r ≥ 0, if [Formula: see text] is zero for all 𝔭 in Supp (M), then [Formula: see text] for all i ≥ r.

Cofiniteness and vanishing of local cohomology modules

1991 ◽  
Vol 110 (3) ◽  
pp. 421-429 ◽  
Author(s):  
Craig Huneke ◽  
Jee Koh
Keyword(s):  
Local Ring ◽  
Maximal Ideal ◽  
Local Cohomology ◽  
Residue Field ◽  
Finitely Generated ◽  
Noetherian Local Ring ◽  
Local Cohomology Modules ◽  
Cohomology Modules

Let R be a noetherian local ring with maximal ideal m and residue field k. If M is a finitely generated R-module then the local cohomology modules are known to be Artinian. Grothendieck [3], exposé 13, 1·2 made the following conjecture:If I is an ideal of R and M is a finitely generated R-module, then HomR (R/I, ) is finitely generated.

scholarly journals FROBENIUS ACTIONS ON LOCAL COHOMOLOGY MODULES AND DEFORMATION

2017 ◽  
Vol 232 ◽  
pp. 55-75 ◽  
Author(s):  
LINQUAN MA ◽  
PHAM HUNG QUY
Keyword(s):  
Local Ring ◽  
Maximal Ideal ◽  
Local Cohomology ◽  
Noetherian Local Ring ◽  
Local Cohomology Modules ◽  
Nonzero Divisor ◽  
Cohomology Modules

Let $(R,\mathfrak{m})$ be a Noetherian local ring of characteristic $p>0$. We introduce and study $F$-full and $F$-anti-nilpotent singularities, both are defined in terms of the Frobenius actions on the local cohomology modules of $R$ supported at the maximal ideal. We prove that if $R/(x)$ is $F$-full or $F$-anti-nilpotent for a nonzero divisor $x\in R$, then so is $R$. We use these results to obtain new cases on the deformation of $F$-injectivity.

scholarly journals Notes on endomorphisms, local cohomology and completion

2021 ◽  
pp. 169-180
Author(s):  
Peter Schenzel
Keyword(s):  
Noetherian Ring ◽  
Local Cohomology ◽  
Local Cohomology Module ◽  
Finitely Generated Module ◽  
Content Type ◽  
Local Cohomology Modules ◽  
Derived Functors ◽  
Adic Completion ◽  
Cohomology Modules

Let M M denote a finitely generated module over a Noetherian ring R R . For an ideal I ⊂ R I \subset R there is a study of the endomorphisms of the local cohomology module H I g ( M ) , g = g r a d e ( I , M ) , H^g_I(M), g = grade(I,M), and related results. Another subject is the study of left derived functors of the I I -adic completion Λ i I ( H I g ( M ) ) \Lambda ^I_i(H^g_I(M)) , motivated by a characterization of Gorenstein rings given in [25]. This provides another Cohen-Macaulay criterion. The results are illustrated by several examples. There is also an extension to the case of homomorphisms of two different local cohomology modules.

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