## Kyoto Journal of Mathematics

### On the cofiniteness of generalized local cohomology modules

#### Abstract

Let $R$ be a commutative Noetherian ring, let $I$ be an ideal of $R$, and let $M$, $N$ be two finitely generated $R$-modules. The aim of this paper is to investigate the $I$-cofiniteness of generalized local cohomology modules $H^{j}_{I}(M,N)={\varinjlim}_{n}\operatorname{Ext}^{j}_{R}(M/I^{n}M,N)$ of $M$ and $N$ with respect to $I$. We first prove that if $I$ is a principal ideal, then $H^{j}_{I}(M,N)$ is $I$-cofinite for all $M$, $N$ and all $j$. Secondly, let $t$ be a nonnegative integer such that $\operatorname{dim}\operatorname{Supp}(H^{j}_{I}(M,N))\le1$ for all $j\lt t$. Then $H^{j}_{I}(M,N)$ is $I$-cofinite for all $j\lt t$ and $\operatorname{Hom}(R/I,H^{t}_{I}(M,N))$ is finitely generated. Finally, we show that if $\operatorname{dim}(M)\le2$ or $\operatorname{dim}(N)\le2$, then $H^{j}_{I}(M,N)$ is $I$-cofinite for all $j$.

#### Article information

Source
Kyoto J. Math. Volume 55, Number 1 (2015), 169-185.

Dates
First available in Project Euclid: 13 March 2015

https://projecteuclid.org/euclid.kjm/1426252134

Digital Object Identifier
doi:10.1215/21562261-2848151

Mathematical Reviews number (MathSciNet)
MR3323531

Zentralblatt MATH identifier
1316.13024

#### Citation

Cuong, Nguyen Tu; Goto, Shiro; Van Hoang, Nguyen. On the cofiniteness of generalized local cohomology modules. Kyoto J. Math. 55 (2015), no. 1, 169--185. doi:10.1215/21562261-2848151. https://projecteuclid.org/euclid.kjm/1426252134

#### References

• [1] K. Bahmanpour and R. Naghipour, On the cofiniteness of local cohomology modules, Proc. Amer. Math. Soc. 136 (2008), 2359–2363.
• [2] K. Bahmanpour and R. Naghipour, Cofiniteness of local cohomology modules for ideals of small dimension, J. Algebra 321 (2009), 1997–2011.
• [3] M. H. Bijan-Zadeh, A common generalization of local cohomology theories, Glasgow Math. J. 21 (1980), 173–181.
• [4] K. Borna, P. Sahandi, and S. Yassemi, Cofiniteness of generalized local cohomology modules, Bull. Aust. Math. Soc. 83 (2011), 382–388.
• [5] M. Brodmann and A. L. Faghani, A finiteness result for associated primes of local cohomology modules, Proc. Amer. Math. Soc. 128 (2000), 2851–2853.
• [6] M. Brodmann and R. Y. Sharp, Local Cohomology: An Algebraic Introduction with Geometric Applications, Cambridge Stud. Adv. Math 60, Cambridge Univ. Press, Cambridge, 1998.
• [7] M. Chardin and K. Divaani-Aazar, A duality theorem for generalized local cohomology, Proc. Amer. Math. Soc. 136 (2008), 2749–2754.
• [8] M. Chardin and K. Divaani-Aazar, Generalized local cohomology and regularity of Ext modules, J. Algebra 319 (2008), 4780–4797.
• [9] N. T. Cuong and N. V. Hoang, Some finite properties of generalized local cohomology modules, East-West J. Math. 7 (2005), 107–115.
• [10] N. T. Cuong and N. V. Hoang, On the vanishing and the finiteness of supports of generalized local cohomology modules, Manuscripta Math. 126 (2008), 59–72.
• [11] N. T. Cuong and L. T. Nhan, On the Noetherian dimension of Artinian modules, Vietnam J. Math. 30 (2002), 121–130.
• [12] D. Delfino and T. Marley, Cofinite modules and local cohomology, J. Pure Appl. Algebra 121 (1997), 45–52.
• [13] K. Divaani-Aazar and A. Mafi, Associated primes of local cohomology modules, Proc. Amer. Math. Soc. 133 (2005), 655–660.
• [14] K. Divaani-Aazar and R. Sazeedeh, Cofiniteness of generalized local cohomology modules, Colloq. Math. 99 (2004), 283–290.
• [15] A. Grothendieck, Cohomologie local des faisceaux coherents et théorèmes de Lefschetz locaux et globaux, Séminaire de Géométrie Algébrique du Bois Marie 1962 (SGA 2), North-Holland, Amsterdam, 1968.
• [16] R. Hartshorne, Affine duality and cofiniteness, Invent. Math. 9 (1969/1970), 145–164.
• [17] J. Herzog, Komplexe, Auflösungen und Dualität in der Lokalen Algebra, Habilitationsschrift, Universität Regensburg, 1970.
• [18] J. Herzog and N. Zamani, Duality and vanishing of generalized local cohomology, Arch. Math. (Basel) 81 (2003), 512–519.
• [19] N. V. Hoang, On the associated primes and the supports of generalized local cohomology modules, Acta Math. Vietnam 33 (2008), 163–171.
• [20] S. Kawakami and K. I. Kawasaki, On the finiteness of Bass numbers of generalized local cohomology modules, Toyama Math. J. 29 (2006), 59–64.
• [21] K. I. Kawasaki, On the finiteness of Bass numbers of local cohomology modules, Proc. Amer. Math. Soc. 124 (1996), 3275–3279.
• [22] K. I. Kawasaki, Cofiniteness of local cohomology modules for principal ideals, Bull. London Math. Soc. 30 (1998), 241–246.
• [23] K. I. Kawasaki, On a category of cofinite modules which is Abelian, Math. Z. 269 (2011), 587–608.
• [24] I. G. Macdonald, “Secondary representation of modules over a commutative ring” in Symposia Mathematica, Vol. XI (Rome, 1971), Academic Pres, London, 1973, 23–43.
• [25] T. Marley, Associated primes of local cohomology module over rings of small dimension, Manuscripta Math. 104 (2001), 519–525.
• [26] T. Marley and J. C. Vassilev, Cofiniteness and associated primes of local cohomology modules, J. Algebra 256 (2002), 180–193.
• [27] L. Melkersson, On asymptotic stability for sets of prime ideals connected with the powers of an ideal, Math. Proc. Cambridge Philos. Soc. 107 (1990), 267–271.
• [28] L. Melkersson, Some applications of a criterion for Artinianness of a module, J. Pure Appl. Algebra 101 (1995), 291–303.
• [29] L. Melkersson, Properties of cofinite modules and applications to local cohomology, Math. Proc. Cambridge Phil. Soc. 125 (1999), 417–423.
• [30] L. Melkersson, Modules cofinite with respect to an ideal, J. Algebra 285 (2005), 649–668.
• [31] N. Suzuki, On the generalized local cohomology and its duality, J. Math. Kyoto Univ. 18 (1978), 71–85.
• [32] S. Yassemi, Generalized section functors, J. Pure Appl. Algebra 95 (1994), 103–119.
• [33] K. I. Yoshida, Cofiniteness of local cohomology modules for ideals of dimension one, Nagoya Math. J. 147 (1997), 179–191.
• [34] H. Zöschinger, Minimax moduln, J. Algebra 102 (1986), 1–32.