Local-global principle for annihilation of general local cohomology

2001 ◽  
Vol 87 (1) ◽  
pp. 129-136 ◽  
Author(s):  
J. Asadollahi ◽  
K. Khashyarmanesh ◽  
Sh. Salarian
Keyword(s):  
Local Cohomology ◽  
Global Principle

scholarly journals A local-global principle for small triangulated categories

2015 ◽  
Vol 158 (3) ◽  
pp. 451-476 ◽  
Author(s):  
DAVE BENSON ◽  
SRIKANTH B. IYENGAR ◽  
HENNING KRAUSE
Keyword(s):  
Noetherian Ring ◽  
Local Cohomology ◽  
Triangulated Category ◽  
Triangulated Categories ◽  
Commutative Noetherian Ring ◽  
Global Principle ◽  
Compactly Generated

AbstractLocal cohomology functors are constructed for the category of cohomological functors on an essentially small triangulated category ⊺ equipped with an action of a commutative noetherian ring. This is used to establish a local-global principle and to develop a notion of stratification, for ⊺ and the cohomological functors on it, analogous to such concepts for compactly generated triangulated categories.

Remarks on the local-global principle for a subcategory consisting of extension modules

2019 ◽  
Vol 18 (12) ◽  
pp. 1950236
Author(s):  
Takeshi Yoshizawa
Keyword(s):  
Noetherian Ring ◽  
Local Cohomology ◽  
Finitely Generated ◽  
Commutative Noetherian Ring ◽  
Local Cohomology Modules ◽  
Global Principles ◽  
Finitely Generated Modules ◽  
Cohomology Modules ◽  
Global Principle

Faltings presented the local-global principle for the finiteness dimension of local cohomology modules. This paper deals with the local-global principle for an extension subcategory over a commutative Noetherian ring. We prove that finitely generated modules satisfy the local-global principles for certain extension subcategories. Additionally, we provide a generalization of Faltings’ local-global principle, which also includes the local-global principles for the Artinianness and Minimaxness of local cohomology modules.

scholarly journals The derived category analogues of Faltings Local-global Principle and Annihilator Theorems

2019 ◽  
Vol 18 (07) ◽  
pp. 1950140 ◽  
Author(s):  
Kamran Divaani-Aazar ◽  
Majid Rahro Zargar
Keyword(s):  
Closed Subset ◽  
Local Cohomology ◽  
Derived Category ◽  
Finitely Generated ◽  
Local Cohomology Modules ◽  
Bounded Complex ◽  
Cohomology Modules ◽  
Global Principle

Let [Formula: see text] be a specialization closed subset of Spec R and X a homologically left-bounded complex with finitely generated homologies. We establish Faltings’ Local-global Principle and Annihilator Theorems for the local cohomology modules [Formula: see text] Our versions contain variations of results already known on these theorems.

Faltings’ local–global principle for the in dimension

2018 ◽  
Vol 46 (8) ◽  
pp. 3496-3509 ◽  
Author(s):  
Reza Naghipour ◽  
Robabeh Maddahali ◽  
Khadijeh Ahmadi Amoli
Keyword(s):  
Local Cohomology ◽  
Local Cohomology Modules ◽  
Cohomology Modules ◽  
Global Principle

scholarly journals LOCAL-GLOBAL PRINCIPLE FOR THE FINITENESS AND ARTINIANNESS OF GENERALISED LOCAL COHOMOLOGY MODULES

2012 ◽  
Vol 87 (3) ◽  
pp. 480-492
Author(s):  
ALI FATHI
Keyword(s):  
Prime Ideal ◽  
Local Cohomology ◽  
Affirmative Answer ◽  
Fixed Integer ◽  
Commutative Noetherian Ring ◽  
Serre Subcategory ◽  
Local Cohomology Modules ◽  
Global Principles ◽  
Cohomology Modules ◽  
Global Principle

AbstractLet $\mathcal S$ be a Serre subcategory of the category of $R$-modules, where $R$ is a commutative Noetherian ring. Let $\mathfrak a$ and $\mathfrak b$ be ideals of $R$ and let $M$ and $N$ be finite $R$-modules. We prove that if $N$ and $H^i_{\mathfrak a}(M,N)$ belong to $\mathcal S$ for all $i\lt n$ and if $n\leq \mathrm {f}$-$\mathrm {grad}({\mathfrak a},{\mathfrak b},N )$, then $\mathrm {Hom}_{R}(R/{\mathfrak b},H^n_{{\mathfrak a}}(M,N))\in \mathcal S$. We deduce that if either $H^i_{\mathfrak a}(M,N)$ is finite or $\mathrm {Supp}\,H^i_{\mathfrak a}(M,N)$ is finite for all $i\lt n$, then $\mathrm {Ass}\,H^n_{\mathfrak a}(M,N)$ is finite. Next we give an affirmative answer, in certain cases, to the following question. If, for each prime ideal ${\mathfrak {p}}$ of $R$, there exists an integer $n_{\mathfrak {p}}$ such that $\mathfrak b^{n_{\mathfrak {p}}} H^i_{\mathfrak a R_{\mathfrak {p}}}({M_{\mathfrak {p}}},{N_{\mathfrak {p}}})=0$ for every $i$ less than a fixed integer $t$, then does there exist an integer $n$ such that $\mathfrak b^nH^i_{\mathfrak a}(M,N)=0$ for all $i\lt t$? A formulation of this question is referred to as the local-global principle for the annihilation of generalised local cohomology modules. Finally, we prove that there are local-global principles for the finiteness and Artinianness of generalised local cohomology modules.

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