scholarly journals Local cohomology in Grothendieck categories

2019 ◽  
Vol 19 (11) ◽  
pp. 2050222
Author(s):  
Fatemeh Savoji ◽  
Reza Sazeedeh
Keyword(s):  
Open Subset ◽  
Local Cohomology ◽  
Derived Category ◽  
Cohomology Theory ◽  
Grothendieck Category

Let [Formula: see text] be a locally noetherian Grothendieck category. In this paper, we define and study the section functor on [Formula: see text] with respect to an open subset of [Formula: see text]. Next, we define and study local cohomology theory in [Formula: see text] in terms of the section functors. Finally, we study abstract local cohomology functor on the derived category [Formula: see text].

scholarly journals On the cominimaxness of generalized local cohomology modules

2020 ◽  
Vol 23 (1) ◽  
pp. 479-483
Author(s):  
Cam Thi Hong Bui ◽  
Tri Minh Nguyen
Keyword(s):  
Algebraic Geometry ◽  
Commutative Algebra ◽  
Local Cohomology ◽  
Cohomology Theory ◽  
Negative Integer ◽  
Finitely Generated ◽  
Local Cohomology Module ◽  
Local Cohomology Modules ◽  
Generalized Local Cohomology Modules ◽  
Cohomology Modules

The local cohomology theory plays an important role in commutative algebra and algebraic geometry. The I-cofiniteness of local cohomology modules is one of interesting properties which has been studied by many mathematicians. The I-cominimax modules is an extension of I-cofinite modules which was introduced by Hartshorne. An R-module M is I-cominimax if Supp_R(M)\subseteq V(I) and Ext^i_R(R/I,M) is minimax for all  i\ge 0. In this paper, we show some conditions such that the generalized local cohomology module H^i_I(M,N) is I-cominimax for all i\ge 0. We show that if H^i_I(M,K) is I-cofinite for all i<t and all finitely generated R-module K, then  H^i_I(M,N) is I-cominimax for all i<t  and all minimax R-module N.  If M is a finitely generated R-module, N is a minimax R-module and t is a non-negative integer such that  dim Supp_R(H^i_I(M,N))\le 1 for all i<t then H^i_I(M,N) is I-cominimax for all  i<t. When  dim R/I\le 1 and H^i_I(N) is I-cominimax for all  i\ge 0 then H^i_I(M,N) is I-cominimax for all i\ge 0.

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.

Generalized local homology and duality

2019 ◽  
Vol 29 (03) ◽  
pp. 581-601
Author(s):  
Do Ngoc Yen ◽  
Tran Tuan Nam
Keyword(s):  
Local Cohomology ◽  
Homology Theory ◽  
Cohomology Theory ◽  
Local Homology

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.

scholarly journals A generalization of Hartshorne's connectedness theorem

2022 ◽  
Vol 40 ◽  
pp. 1-5
Author(s):  
Dawood Hassanzadeh-lelekaami
Keyword(s):  
Local Cohomology ◽  
Cohomology Theory ◽  
Prime Spectrum ◽  
Connectedness Property

In this paper, we use local cohomology theory to present some results about connectedness property of prime spectrum of modules. In particular, we generalize the Hartshorne's connectedness theorem.

scholarly journals Pure exact structures and the pure derived category of a scheme

2016 ◽  
Vol 163 (2) ◽  
pp. 251-264 ◽  
Author(s):  
SERGIO ESTRADA ◽  
JAMES GILLESPIE ◽  
SINEM ODABAŞI
Keyword(s):  
Derived Category ◽  
Category Structure ◽  
Model Category ◽  
Grothendieck Category ◽  
Coherent Sheaves ◽  
Monoidal Structure ◽  
Chain Complexes

AbstractLet$\mathcal{C}$be closed symmetric monoidal Grothendieck category. We define the pure derived category with respect to the monoidal structure via a relative injective model category structure on the categoryC($\mathcal{C}$) of unbounded chain complexes in$\mathcal{C}$. We use λ-Purity techniques to get this. As application we define the stalkwise pure derived category of the category of quasi–coherent sheaves on a quasi-separated scheme. We also give a different approach by using the category of flat quasi–coherent sheaves.

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