Cofiniteness of general local cohomology modules for small dimensions
1
조회
0
다운로드
0
인용
0
참고문헌
키워드
초록
Let $R$ be a commutative Noetherian ring, $\Phi$ a system of ideals of $R$ and $I\in \Phi$. In this paper among other things we prove that if $M$ is finitely generated and $t\in \mathbb{N}$ such that the $R$-module $\lc^{i}_\Phi(M)$ is ${\rm FD_{\leq 1}}$ (or weakly Laskerian) for all $i<t$, then $\lc^{i}_\Phi(M)$ is $\Phi$-cofinite for all $i<t$ and for any ${\rm FD_{\leq 0}}$ (or minimax) submodule $N$ of $\lc^t_\Phi(M)$, the $R$-modules $\Hom_R(R/I,\lc^t_\Phi(M)/N)$ and $\Ext^1_R(R/I,\lc^t_\Phi(M)/N)$ are finitely generated. Also it is shown that if $\cd I = 1$ or $\dim M/IM \leq 1$ {\rm(}e.g., $\dim R/I \leq 1${\rm)} for all $I \in \Phi$, then the local cohomology module $\lc^{i}_\Phi(M)$ is $\Phi$-cofinite for all $i\geq 0$. These generalize the main results of Aghapournahr and Bahmanpour \cite{AB}, Bahmanpour and Naghipour \cite{BN2, BN}. Also we study cominimaxness and weakly cofiniteness of local cohomology modules with respect to a system of ideals.
Abstract
Let $R$ be a commutative Noetherian ring, $\Phi$ a system of ideals of $R$ and $I\in \Phi$. In this paper among other things we prove that if $M$ is finitely generated and $t\in \mathbb{N}$ such that the $R$-module $\lc^{i}_\Phi(M)$ is ${\rm FD_{\leq 1}}$ (or weakly Laskerian) for all $i<t$, then $\lc^{i}_\Phi(M)$ is $\Phi$-cofinite for all $i<t$ and for any ${\rm FD_{\leq 0}}$ (or minimax) submodule $N$ of $\lc^t_\Phi(M)$, the $R$-modules $\Hom_R(R/I,\lc^t_\Phi(M)/N)$ and $\Ext^1_R(R/I,\lc^t_\Phi(M)/N)$ are finitely generated. Also it is shown that if $\cd I = 1$ or $\dim M/IM \leq 1$ {\rm(}e.g., $\dim R/I \leq 1${\rm)} for all $I \in \Phi$, then the local cohomology module $\lc^{i}_\Phi(M)$ is $\Phi$-cofinite for all $i\geq 0$. These generalize the main results of Aghapournahr and Bahmanpour \cite{AB}, Bahmanpour and Naghipour \cite{BN2, BN}. Also we study cominimaxness and weakly cofiniteness of local cohomology modules with respect to a system of ideals.
참고문헌 (33)
- M. Aghapournahr, "Cofiniteness of certain local cohomology modules for small dimen-sions", preprint, ~
- M. Aghapournahr, "Cofiniteness of weakly Laskerian local coho-mology modules", Bull. Math. Soc. Sci. Math. Roumanie (N.S), 57(105)(4), pp.347~356 (2014)
- M. Aghapournahr, "A natural map in local cohomology", Ark. Mat, 48(2), pp.243~251 (2010)
- J. Asadollahi, "A generalization of the cofiniteness problem in local cohomology modules", J. Aust. Math. Soc, 75(3), pp.313~324 (2003)
- K. Bahmanpour, "On the category of weakly Laskerian cofinite modules", Math. Scand, 115(1), pp.62~68 (2014)
- K. Bahmanpour, "On the cofiniteness of local cohomology modules", Proc. Amer. Math. Soc, 136(7), pp.2359~2363 (2008)
- K. Bahmanpour, "Cofiniteness of local cohomology modules for ideals of small dimension", J. Algebra, 321(7), pp.1997~2011 (2009)
- K. Bahmanpour, "On the category of cofinite modules which is Abelian", Proc. Amer. Math. Soc, 142(4), pp.1101~1107 (2014)
- R. Belshoff, "The local cohomology modules of Matlis reflexive modules are almost cofinite", Proc. Amer. Math. Soc, 124(9), pp.2649~2654 (1996)
- R. Belshoff, "Finiteness properties of Matlis reflexive modules", Comm. Algebra, 24(4), pp.1371~1376 (1996)
- M. H. Bijan-Zadeh, "Torsion theories and local cohomology over commutative Noetherian ring", J. Lond. Math. Soc. (2), 19(3), pp.402~410 (1979)
- M. H. Bijan-Zadeh, "A common generalization of local cohomology theories", Glasgow Math. J, 21(2), pp.173~181 (1980)
- M. H. Bijan-Zadeh, "On the Artinian property of certain general local cohomology", J. Lond. Math. Soc. (2), 35(3), pp.399~403 (1985)
- M. P. Brodmann, "A finiteness result for associated primes of local cohomology modules", Proc. Amer. Math. Soc, 128(10), pp.2851~2853 (2000)
- M. P. Brodmann, "Local Cohomology: An algebraic introduction with geometric applications" (1998), Cambridge. Univ. Press
- W. Bruns, "Cohen Macaulay Rings" (1993), Cambridge Univ. Press
- D. Delfino, "Cofinite modules and local cohomology", J. Pure Appl. Algebra, 121(1), pp.45~52 (1997)
- M. T. Dibaei, "Associated primes and cofiniteness of local cohomology modules", Manuscripta Math, 117(2), pp.199~205 (2005)
- M. T. Dibaei, "Abelian groups, rings, modules and homological algebra", pp.2006, Chapman and Hall
- K. Divaani-Aazar, "Associated primes of local cohomology modules", Proc. Amer. Math. Soc, 133(3), pp.655~660 (2005)
- A. Grothendieck, "Cohomologie locale des faisceaux coh´erents et theoremes de Lefschetz locaux et globaux (SGA2)" (1968), North-Holland
- R. Hartshorne, "Affine duality and cofiniteness", Invent. Math, 9, pp.145~164 (1970)
- P. Hung Quy, "On the finiteness of associated primes of local cohomology modules", Proc. Amer. Math. Soc, 138(6), pp.1965~1968 (2010)
- K. Khashyarmanesh, "On the associated primes of local cohomology modules", Comm. Algebra, 27(12), pp.6191~6198 (1999)
- A. Mafi, "On the cofiniteness properties of certain general local cohomology modules", Acta Sci. Math. (Szeged), 74(3-4), pp.501~507 (2008)
- T. Marley, "The associated primes of local cohomology modules over rings of small di-mension", Manuscripta Math, 104(4), pp.519~525 (2001)
- T. Marley, "Cofiniteness and associated primes of local cohomology modules", J. Algebra, 256(1), pp.180~193 (2002)
- H. Matsumura, "Commutative Ring Theory" (1986), Cambridge Univ. Press
- L. Melkersson, "Some applications of a criterion for artinianness of a module", J. Pure Appl. Algebra, 101(3), pp.291~303 (1995)
- L. Melkersson, "Properties of cofinite modules and applications to local cohomology", Math. Proc. Cambridge Philos. Soc, 125(3), pp.417~423 (1999)
- L. Melkersson, "Modules cofinite with respect to an ideal", J. Algebra, 285(2), pp.649~668 (2005)
- T. Yoshizawa, "Subcategories of extension modules by Serre subcategories", Proc. Amer. Math. Soc, 140(7), pp.2293~2305 (2012)
- H. Zoschinger, "Minimax Moduln", J. Algebra, 102(1), pp.1~32 (1986)