Cominimaxness of certain general local cohomology modules

Yıl 2019, Cilt: 48 Sayı: 4, 1121 - 1130, 08.08.2019

Moharram Aghapournahr

Öz

Let $R$ be a commutative Noetherian ring, $\Phi$ a system of ideals of $R$ and $I\in \Phi$. Let $t\in\Bbb{N}_0$ be an integer and $M$ an $R$-module such that $Ext^i_R(R/I,M)$ is minimax for all $i\leq t+1$. We prove that if the $R$-module $H^{i}_\Phi(M)$ is ${FD_{\leq 1}}$ (or weakly Laskerian) for all $i<t$, then $H^i_{\Phi}(M)$ is $\Phi$-cominimax for all $i<t$ and for any $FD_{\leq 0}$ (or minimax) submodule $N$ of $H^t_{\Phi}(M)$, the $R$-modules $Hom_{R}(R/I,H^t_{\Phi}(M)/N)$ and $Ext^1_{R}(R/I,H^t_{\Phi}(M)/N)$ are minimax. Let $N$ be a finitely generated $R$-module. We also prove that $Ext^j_{R}(N,H^i_\Phi(M))$ and $Tor^R_{j}(N,H^i_{\Phi}(M))$ are $\Phi$-cominimax for all $i$ and $j$ whenever $M$ is minimax and $H^i_{\Phi}(M)$ is ${FD_{\leq 1}}$ (or weakly Laskerian) for all $i$.

Anahtar Kelimeler

Local cohomology, ${\rm FD_{\leq n}}$ modules, $ETH$-cominimax modules

Kaynakça

• [1] A. Abbasi, D. Hasanzadeh-Lelekami and H. Roshan-Shokalgourabi, Some results on the local cohomology of minimax modules, Czechoslovak Math. J. 64 (2), 327–333, 2014.
• [2] M. Aghapournahr, On cofiniteness of local cohomology modules for a pair of ideals for small dimensions, J. Algebra Appl. 17 (2), 1850020, 2018.
• [3] M. Aghapournahr, Cominimaxness of local cohomology modules, Czechoslovak Math. J. 69 (1), 75–86, 2019.
• [4] M. Aghapournahr and K. Bahmanpour, Cofiniteness of weakly Laskerian local cohomology modules, Bull. Math. Soc. Sci. Math. Roumanie 105 (4), 347–356, 2014.
• [5] M. Aghapournahr and K. Bahmanpour, Cofiniteness of general local cohomology modules for small dimensions, Bull. Korean Math. Soc. 53 (5), 1341–1352, 2016.
• [6] M. Aghapournahr, L. Melkersson, A natural map in local cohomology, Ark. Mat. 48 (2), 243–251, 2010.
• [7] J. Asadollahi, K. Khashyarmanesh and Sh. Salarian, A generalization of the cofiniteness problem in local cohomology modules, J. Aust. Math. Soc. 75 (3), 313–324, 2003.
• [8] J. Azami, R. Naghipour, and B. Vakili, Finiteness properties of local cohomology modules for a-minimax modules, Proc. Amer. Math. Soc. 137 (2), 439–448, 2009.
• [9] K. Bahmanpour and R. Naghipour, On the cofiniteness of local cohomology modules, Proc. Amer. Math. Soc. 136 (2008), 2359–2363.
• [10] K. Bahmanpour and R. Naghipour, Cofiniteness of local cohomology modules for ideals of small dimension, J. Algebra 321 (7), 1997–2011, 2009.
• [11] K. Bahmanpour, R. Naghipour and M. Sedghi, On the category of cofinite modules which is Abelian, Proc. Amer. Math. Soc. 142 (4), 1101–1107, 2014.
• [12] M.H. Bijan-Zadeh, Torsion theory and local cohomology over commutative Noetherian ring, J. London Math. Soc. 19 (3), 402–410, 1979.
• [13] M.H. Bijan-Zadeh, A common generalization of local cohomology theories, Glasgow Math. J. 21 (1), 173–181, 1980.
• [14] K. Borna Lorestani, P. Sahandi and S. Yassemi, Artinian local cohomology modules, Canad. Math. Bull. 50 (4), 598–602, 2007.
• [15] M.P. Brodmann and R.Y. Sharp Local cohomology-An algebraic introduction with geometric applications, Cambridge. Univ. Press, 1998.
• [16] W. Bruns and J. Herzog, Cohen Macaulay Rings, in: Cambridge Studies in Advanced Mathematics, 39, Cambridge Univ. Press, Cambridge, UK, 1993.
• [17] D. Delfino and T. Marley, Cofinite modules and local cohomology, J. Pure Appl. Algebra 121 (1), 45–52, 1997.
• [18] M.T. Dibaei and S. Yassemi, Associated primes and cofiniteness of local cohomology modules, Manuscripta Math. 117 (2), 199–205, 2005.
• [19] M.T. Dibaei and S. Yassemi, Associated primes of the local cohomology modules, in: Abelian groups, rings, modules and homological algebra, 49–56, Chapman and Hall/CRC, 2006.
• [20] K. Divaani-Aazar and A. Mafi, Associated primes of local cohomology modules, Proc. Amer. Math. Soc. 133 (3), 655–660, 2005.
• [21] E. Enochs, Flat covers and flat cotorsion modules, Proc. Amer. Math. Soc. 92 (2), 179–184, 1984.
• [22] A. Grothendieck, Cohomologie locale des faisceaux cohérents et théorèmes de Lefschetz locaux et globaux (SGA2), North-Holland, Amsterdam, 1968.
• [23] R. Hartshorne, Affine duality and cofiniteness, Invent. Math. 9 (2), 145–164, 1970.
• [24] D. Hasanzadeh-Lelekami, H. Roshan-Shokalgourabi, Extension functors of cominimax modules, Comm. Algebra 45 (2), 621–629, 2017.
• [25] C. Huneke and J. Koh, Cofiniteness and vanishing of local cohomology modules, Math. Proc. Cambridge Philos. Soc. 110 (3), 421–429, 1991.
• [26] P. Hung Quy, On the finiteness of associated primes of local cohomology modules, Proc. Amer. Math. Soc. 138 (6), 1965–1968, 2010.
• [27] Y. Irani, Cominimaxness with respect to ideals of dimension one, Bull. Korean Math. Soc. 54 (1), 289–298, 2017.
• [28] K.I. Kawasaki, On a category of cofinite modules which is Abelian, Math. Z. 269 (1-2), 587–608, 2011.
• [29] A. Mafi, On the local cohomology of minimax modules, Bull. Korean Math. Soc. 48 (6), 1125–1128, 2011.
• [30] T. Marley and J.C. Vassilev, Cofiniteness and associated primes of local cohomology modules, J. Algebra 256 (1), 180–193, 2002.
• [31] L. Melkersson, Properties of cofinite modules and applications to local cohomology, Math. Proc. Cambridge Philos. Soc. 125 (3), 417–423, 1999.
• [32] L. Melkersson, Modules cofinite with respect to an ideal, J. Algebra 285 (2), 649–668, 2005.
• [33] L. Melkersson, Cofiniteness with respect to ideals of dimension one, J. Algebra 372, 459–462, 2012.
• [34] R. Takahashi, Y. Yoshino and T. Yoshizawa, Local cohomology based on a nonclosed support defined by a pair of ideals, J. Pure Appl. Algebra 213 (4), 582–600, 2009.
• [35] T. Yoshizawa, Subcategories of extension modules by subcategories, Proc. Amer. Math. Soc. 140 (7), 2293–2305, 2012.
• [36] T. Zink, Endlichkeitsbedingungen für Moduln über einem Noetherschen Ring, Math. Nachr. 64 (1), 239–252, 1974.
• [37] H. Zöschinger, Minimax Moduln, J. Algebra 102 (1), 1–32, 1986.