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Search: MSC category 13D45 ( Local cohomology [See also 14B15] )

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1. CMB 2014 (vol 57 pp. 477)

Eghbali, Majid
 On Set Theoretically and Cohomologically Complete Intersection Ideals Let $(R,\mathfrak m)$ be a local ring and $\mathfrak a$ be an ideal of $R$. The inequalities $\operatorname{ht}(\mathfrak a) \leq \operatorname{cd}(\mathfrak a,R) \leq \operatorname{ara}(\mathfrak a) \leq l(\mathfrak a) \leq \mu(\mathfrak a)$ are known. It is an interesting and long-standing problem to find out the cases giving equality. Thanks to the formal grade we give conditions in which the above inequalities become equalities. Keywords:set-theoretically and cohomologically complete intersection ideals, analytic spread, monomials, formal grade, depth of powers of idealsCategories:13D45, 13C14

2. CMB 2012 (vol 56 pp. 491)

Bahmanpour, Kamal
 A Note on Homological Dimensions of Artinian Local Cohomology Modules Let $(R,{\frak m})$ be a non-zero commutative Noetherian local ring (with identity), $M$ be a non-zero finitely generated $R$-module. In this paper for any ${\frak p}\in {\rm Spec}(R)$ we show that $\operatorname{{\rm injdim_{_{R_{\frak p}}}}} H^{i-\dim(R/{\frak p})}_{{\frak p}R_{\frak p}}(M_{\frak p})$ and ${\rm fd}_{R_{\p}} H^{i-\dim(R/{\frak p})}_{{\frak p}R_{\frak p}}(M_{\frak p})$ are bounded from above by $\operatorname{{\rm injdim_{_{R}}}} H^i_{\frak m}(M)$ and ${\rm fd}_R H^i_{\frak m}(M)$ respectively, for all integers $i\geq \dim(R/{\frak p})$. Keywords:cofinite modules, flat dimension, injective dimension, Krull dimension, local cohomologyCategory:13D45

3. CMB 2011 (vol 55 pp. 315)

Hellus, M.
 A Note on the Vanishing of Certain Local Cohomology Modules For a finite module $M$ over a local, equicharacteristic ring $(R,m)$, we show that the well-known formula $\textrm{cd}(m,M)=\dim M$ becomes trivial if ones uses Matlis duals of local cohomology modules together with spectral sequences. We also prove a new ring-theoretic vanishing criterion for local cohomology modules. Category:13D45

4. CMB 2011 (vol 55 pp. 81)

Divaani-Aazar, Kamran; Hajikarimi, Alireza
 Cofiniteness of Generalized Local Cohomology Modules for One-Dimensional Ideals Let $\mathfrak a$ be an ideal of a commutative Noetherian ring $R$ and $M$ and $N$ two finitely generated $R$-modules. Our main result asserts that if $\dim R/\mathfrak a\leq 1$, then all generalized local cohomology modules $H^i_{\mathfrak a}(M,N)$ are $\mathfrak a$-cofinite. Keywords:cofinite modules, generalized local cohomology modules, local cohomology modulesCategories:13D45, 13E05, 13E10

5. CMB 2011 (vol 55 pp. 153)

Mafi, Amir; Saremi, Hero
 Artinianness of Certain Graded Local Cohomology Modules We show that if $R=\bigoplus_{n\in\mathbb{N}_0}R_n$ is a Noetherian homogeneous ring with local base ring $(R_0,\mathfrak{m}_0)$, irrelevant ideal $R_+$, and $M$ a finitely generated graded $R$-module, then $H_{\mathfrak{m}_0R}^j(H_{R_+}^t(M))$ is Artinian for $j=0,1$ where $t=\inf\{i\in{\mathbb{N}_0}: H_{R_+}^i(M)$ is not finitely generated $\}$. Also, we prove that if $\operatorname{cd}(R_+,M)=2$, then for each $i\in\mathbb{N}_0$, $H_{\mathfrak{m}_0R}^i(H_{R_+}^2(M))$ is Artinian if and only if $H_{\mathfrak{m}_0R}^{i+2}(H_{R_+}^1(M))$ is Artinian, where $\operatorname{cd}(R_+,M)$ is the cohomological dimension of $M$ with respect to $R_+$. This improves some results of R. Sazeedeh. Keywords:graded local cohomology, Artinian modulesCategories:13D45, 13E10

6. CMB 2011 (vol 54 pp. 619)

Dibaei, Mohammad T.; Vahidi, Alireza
 Artinian and Non-Artinian Local Cohomology Modules Let $M$ be a finite module over a commutative noetherian ring $R$. For ideals $\mathfrak{a}$ and $\mathfrak{b}$ of $R$, the relations between cohomological dimensions of $M$ with respect to $\mathfrak{a}, \mathfrak{b}$, $\mathfrak{a}\cap\mathfrak{b}$ and $\mathfrak{a}+ \mathfrak{b}$ are studied. When $R$ is local, it is shown that $M$ is generalized Cohen-Macaulay if there exists an ideal $\mathfrak{a}$ such that all local cohomology modules of $M$ with respect to $\mathfrak{a}$ have finite lengths. Also, when $r$ is an integer such that $0\leq r< \dim_R(M)$, any maximal element $\mathfrak{q}$ of the non-empty set of ideals $\{\mathfrak{a} : \textrm{H}_\mathfrak{a}^i(M)$ is not artinian for some $i, i\geq r \}$ is a prime ideal, and all Bass numbers of $\textrm{H}_\mathfrak{q}^i(M)$ are finite for all $i\geq r$. Keywords:local cohomology modules, cohomological dimensions, Bass numbersCategories:13D45, 13E10

7. CMB 2010 (vol 53 pp. 667)

Khashyarmanesh, Kazem
 On the Endomorphism Rings of Local Cohomology Modules Let $R$ be a commutative Noetherian ring and $\mathfrak{a}$ a proper ideal of $R$. We show that if $n:=\operatorname{grade}_R\mathfrak{a}$, then $\operatorname{End}_R(H^n_\mathfrak{a}(R))\cong \operatorname{Ext}_R^n(H^n_\mathfrak{a}(R),R)$. We also prove that, for a nonnegative integer $n$ such that $H^i_\mathfrak{a}(R)=0$ for every $i\neq n$, if $\operatorname{Ext}_R^i(R_z,R)=0$ for all $i >0$ and $z \in \mathfrak{a}$, then $\operatorname{End}_R(H^n_\mathfrak{a}(R))$ is a homomorphic image of $R$, where $R_z$ is the ring of fractions of $R$ with respect to a multiplicatively closed subset $\{z^j \mid j \geqslant 0 \}$ of $R$. Moreover, if $\operatorname{Hom}_R(R_z,R)=0$ for all $z \in \mathfrak{a}$, then $\mu_{H^n_\mathfrak{a}(R)}$ is an isomorphism, where $\mu_{H^n_\mathfrak{a}(R)}$ is the canonical ring homomorphism $R \rightarrow \operatorname{End}_R(H^n_\mathfrak{a}(R))$. Keywords:local cohomology module, endomorphism ring, Matlis dual functor, filter regular sequenceCategories:13D45, 13D07, 13D25

8. CMB 2010 (vol 53 pp. 577)

Asgharzadeh, Mohsen; Tousi, Massoud
 A Unified Approach to Local Cohomology Modules using Serre Classes This paper discusses the connection between the local cohomology modules and the Serre classes of $R$-modules. This connection has provided a common language for expressing some results regarding the local cohomology $R$-modules that have appeared in different papers. Keywords:associated prime ideals, local cohomology modules, Serre classCategory:13D45

9. CMB 2007 (vol 50 pp. 598)

Lorestani, Keivan Borna; Sahandi, Parviz; Yassemi, Siamak
 Artinian Local Cohomology Modules Let $R$ be a commutative Noetherian ring, $\fa$ an ideal of $R$ and $M$ a finitely generated $R$-module. Let $t$ be a non-negative integer. It is known that if the local cohomology module $\H^i_\fa(M)$ is finitely generated for all $i Keywords:local cohomology module, Artinian module, reflexive moduleCategories:13D45, 13E10, 13C05 10. CMB 2002 (vol 45 pp. 119) Marcelo, Agustín; Marcelo, Félix; Rodríguez, César  The Grade Conjecture and the$S_{2}$Condition Sufficient conditions are given in order to prove the lowest unknown case of the grade conjecture. The proof combines vanishing results of local cohomology and the$S_{2}\$ condition. Categories:13D22, 13D45, 13D25, 13C15

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