Finiteness properties of local cohomology modules (an application ofD-modules to commutative algebra)

1993 ◽  
Vol 113 (1) ◽  
pp. 41-55 ◽  
Author(s):  
Gennady Lyubeznik
Keyword(s):  
Commutative Algebra ◽  
Local Cohomology ◽  
Local Cohomology Modules ◽  
Finiteness Properties ◽  
Cohomology Modules

scholarly journals Finiteness properties of generalized local cohomology modules for minimax modules

2017 ◽  
Vol 4 (1) ◽  
pp. 1327683
Author(s):  
Sh. Payrovi ◽  
I. Khalili-Gorji ◽  
Z. Rahimi-Molaei ◽  
Lishan Liu
Keyword(s):  
Local Cohomology ◽  
Local Cohomology Modules ◽  
Minimax Modules ◽  
Generalized Local Cohomology Modules ◽  
Finiteness Properties ◽  
Cohomology Modules

ON THE FINITENESS PROPERTIES OF THE GENERALIZED LOCAL COHOMOLOGY MODULES

2002 ◽  
Vol 30 (2) ◽  
pp. 859-867 ◽  
Author(s):  
J. Asadollahi ◽  
K. Khashyarmanesh ◽  
Sh. Salarian
Keyword(s):  
Local Cohomology ◽  
Local Cohomology Modules ◽  
Generalized Local Cohomology Modules ◽  
Finiteness Properties ◽  
Cohomology Modules

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.

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).

scholarly journals On the Finiteness Properties of Local Cohomology Modules for Regular Local Rings

2017 ◽  
Vol 40 (1) ◽  
pp. 83-96
Author(s):  
Monireh SEDGHI ◽  
Kamal BAHMANPOUR ◽  
Reza NAGHIPOUR
Keyword(s):  
Local Cohomology ◽  
Local Rings ◽  
Local Cohomology Modules ◽  
Regular Local Rings ◽  
Finiteness Properties ◽  
Cohomology Modules

Finiteness properties of local cohomology modules over differentiable admissible algebras

2017 ◽  
Vol 221 (9) ◽  
pp. 2236-2249 ◽  
Author(s):  
L. Alba-Sarria ◽  
R. Callejas-Bedregal ◽  
N. Caro-Tuesta
Keyword(s):  
Local Cohomology ◽  
Local Cohomology Modules ◽  
Finiteness Properties ◽  
Cohomology Modules
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