ANNIHILATORS OF TOP LOCAL COHOMOLOGY MODULES DEFINED BY A PAIR OF IDEALS

Year 2019, , 87 - 94, 11.07.2019

Susan Karimi Shiroyeh Payrovi

https://doi.org/10.24330/ieja.586962

Abstract

Let  $R$ be a commutative Noetherian ring,  $I, J$ two proper ideals of

$R$ and let $M$ be a non-zero finitely generated  $R$-module with $c={\rm cd}(I,J,M)$.

In this paper, we first  introduce $T_R(I,J,M)$ as the largest submodule of $M$

with the property that ${\rm cd}(I,J,T_R(I,J,M))<c$ and we describe it in terms of the reduced primary

decomposition of zero submodule of $M$. It is shown that

 ${\rm Ann}_R(H_{I,J}^d(M))={\rm Ann}_R(M/{T_R(I,J,M)})$ and ${\rm Ann}_R(H_{I}^d(M))={\rm Ann}_R(H_{I,J}^d(M))$,

whenever $R$ is a  local ring, $M$ has dimension $d$ with $H_{I,J}^d(M)\\\neq0$ and

$J^tM\subseteq T_R(I,M)$ for some positive integer $t$.

Keywords

Attached prime ideal, top local cohomology module

References

• A. Atazadeh, M. Sedghi and R. Naghipour, On the annihilators and attached primes of top local cohomology modules, Arch. Math., 102(3) (2014), 225-236.
• K. Bahmanpour, J. A'zami and Gh. Ghasemi, On the annihilators of local cohomology modules, J. Algebra, 363 (2012), 8-13.
• M. P. Brodmann and R. Y. Sharp, Local Cohomology: an Algebraic Introduction with Geometric Applications, Cambridge Studies in Advanced Math- ematics, 60, Cambridge University Press, Cambridge, 1998.
• L. Chu, Top local cohomology modules with respect to a pair of ideals, Proc. Amer. Math. Soc., 139(3) (2011), 777-782.
• L. Chu and W. Wang, Some results on local cohomology modules dened by a pair of ideals, J. Math. Kyoto Univ., 49 (2009), 193-200.
• C. Huneke and J. Koh, Coniteness and vanishing of local cohomology modules, Math. Proc. Cambridge Philos. Soc., 110(3) (1991), 421-429.
• L. R. Lynch, Annihilators of top local cohomology, Comm. Algebra, 40(2) (2012), 542-551.
• G. Lyubeznik, Finiteness properties of local cohomology modules (an application of D-modules to commutative algebra), Invent. Math., 113(1) (1993), 41-55.
• L. T. Nhan and T. D. M. Chau, On the top local cohomology modules, J. Algebra 349 (2012), 342-352.
• Sh. Payrovi and S. Karimi, Upper bounds and attached primes of top local cohomology modules dened by a pair of ideals, J. Hyperstruct., 3(2) (2014), 101-107.
• P. Schenzel, Cohomological annihilators, Math. Proc. Cambridge Philos. Soc. 91(3) (1982), 345-350.
• R. Takahashi, Y. Yoshino and T. Yoshizawa, Local cohomology based on a nonclosed support dened by a pair of ideals, J. Pure Appl. Algebra, 213(4) (2009), 582-600.