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On formal local cohomology modules with respect to a pair of ideals. (English) Zbl 1348.13026
Let $$R$$ be a Noetherian commutative ring with identity. Let $$\mathfrak a$$, $$I$$ and $$J$$ be ideals of $$R$$ and $$M$$ an $$R$$-module. The notion of formal local cohomology was introduced by P. Schenzel [J. Algebra 315, No. 2, 894–923 (2007; Zbl 1131.13018)].
In the present paper, the authors introduce and study two generalizations of the notion of formal local cohomology. They denote these generalizations by $$\mathfrak {F}_{\mathfrak a,I,J}^i(M)$$ and $$\check{\mathfrak {F}}_{\mathfrak a,I,J}^i(M)$$. We bring the definitions of these notions in the sequel.
Let $W(I,J):=\left\{\mathfrak{p}\in \mathrm{Spec }R\mid I^{n} \subseteq \mathfrak{p}+J\text{ for some } n\geq 1\right\}.$ The endofunctor $$\Gamma_{I,J}(-)$$ on the category of $$R$$-modules is defined by setting $\Gamma_{I,J}(M):= \left\{x\in M \mid\mathrm{Supp}_{R}(Rx) \subseteq W(I,J) \right\},$ for an $$R$$-module $$M$$, and $$\Gamma_{I,J}(f):= f|_{\Gamma_{I,J}(M)}$$ for an $$R$$-homomorphism $$f:M\rightarrow N$$. For each integer $$i\geq 0$$, the $$i$$th local cohomology module of $$M$$ with respect to the pair $$(I,J)$$ is defined to be $$H^{i}_{I,J}(M): = R^{i}\Gamma_{I,J}(M)$$. This notion was introduced by R. Takahashi et al. [J. Pure Appl. Algebra 213, No. 4, 582–600 (2009; Zbl 1160.13013)]. Let $$\underline{x}=x_1,\dots ,x_s$$ be a set of generators of $$I$$. In the same paper, the authors introduced the notion, $$\check{C}_{\underline{x},J}$$, the Ćech complex of $$R$$ with respect to $$(I,J)$$.
For each nonnegative integer $$i$$, the authors define $\check{\mathfrak {F}}_{\mathfrak a,I,J}^i(M):=H^i\big(\underset{n}{\varprojlim}\big (\check{C}_{\underline{x},J}\otimes_R\frac{M}{{\mathfrak a}^nM}\big)\big)$ and $\mathfrak {F}_{\mathfrak a,I,J}^i(M):=\underset{n}{\varprojlim} \big(H^{i}_{I,J}\big(\frac{M}{{\mathfrak a}^nM}\big)\big).$ The authors establish several upper and lower vanishing bounds for these cohomology modules. In particular, they show that if $$(R,\mathfrak m)$$ is local, $$M$$ is finitely generated and $$I+J$$ is $$\mathfrak m$$-primary, then $\dim_R(M/(\mathfrak a+J)M)=\sup\{i\in\mathbb{Z}\mid \mathfrak {F}_{\mathfrak a,I,J}^i(M)\neq 0\}.$

MSC:
 13D45 Local cohomology and commutative rings
Full Text:
References:
 [1] M. Aghapournahr, KH. Ahmadi-Amoli and M.Y. Sadegui, The concept of $$(I,J)$$-Cohen-Macaulay modules , Journal of Algebraic Systems, accepted. [2] M. Asgharzadeh and K. Divaani-Aazar, Finiteness properties of formal local cohomology modules and Cohen- Macaulayness , Comm. Algebra 39 (2011), 1082-1103. · Zbl 1237.13034 · doi:10.1080/00927871003610312 · arxiv:0807.5042 [3] N. Bourbaki, Algébre commutative , Hermann, Paris, 1961-1965. [4] M.P. Brodmann and R.Y. Sharp, Local cohomology , An algebraic introduction with geometric applications , Cambridge University Press, Cambridge, 1998. · Zbl 0903.13006 [5] L. Chu and Q. Wang, Some results on local cohomology modules defined by a pair of ideals , J. Math. Kyoto Univ, 49 (2009), 193-200. · Zbl 1174.13024 · www.math.kyoto-u.ac.jp [6] K. Divaani-Aazar, R. Naghipour and M. Tousi, Cohomological dimension of certain algebraic varieties , Proc. Amer. Math. Soc. 130 (2002), 3537-3544. · Zbl 0998.13007 · doi:10.1090/S0002-9939-02-06500-0 [7] K. Divaani-Aazar and P. Schenzel, Ideal topology, local cohomology and connectedness , Math. Proc. Cambr. Philos. Soc. 131 (2001), 211-226. · Zbl 1078.13510 · doi:10.1017/S0305004101005229 [8] M. Eghbali, On formal local cohomology, colocalization and endomorphism ring of top local cohomology modules , Ph.D. thesis, Universitat Halle-Wittenberg, 2011. [9] G. Faltings, Algebraization of some formal vector bundles , Ann. Math. 110 (1979), 501-514. · Zbl 0395.14005 · doi:10.2307/1971235 [10] A. Grothendieck, Local cohomology, Notes by R. Hartshorne , Lect. Notes Math. 20 , Springer, Berlin, 1966. · Zbl 0145.17602 [11] A. Grothendieck and J. Dieudonné, Éléments de géométrie algébrique III, Publ. Math. IHES 11 (1961). [12] J. Herzog, Komplexe , Auflsungen und Dualitt in der lokalen Algebra, Habilitationsschrift, Universität Regensburg, 1970. [13] C. Huneke, Problems on local cohomology , in Free resolutions in commutative algebra and algebraic geometry , Res. Notes Math. 2 (1992), 93-108. · Zbl 0782.13015 [14] S.B. Iyengar, G.J. Leuschke, A. Leykin, C. Miller, E. Miller, A.K. Singh and U. Walther, Twenty-four hours of local cohomology , Grad. Stud. Math. 87 , American Mathematical Society, 2007. · Zbl 1129.13001 [15] A. Kianezhad, A.J. Taherizadeh and A. Tehranian, Formal local cohomology modules and serre subcategories , J. Sci. Kharazmi University 13 (2013), 337-346. [16] A. Mafi, Some results on the local cohomology modules , Arch. Math (Basel) 87 (2006), 211-216. · Zbl 1102.13018 · doi:10.1007/s00013-006-1674-1 · arxiv:math/0512075 [17] —-, Results on formal local cohomology modules , Bull. Malays. Math. Sci. Soc. 36 (2013), 173-177. · Zbl 1262.13029 · math.usm.my [18] C. Peskine and L. Szpiro, Dimension projective finie et cohomologie locale , Publ. Math. I.H.E.S. 42 (1972), 47-119. · Zbl 0268.13008 · doi:10.1007/BF02685877 · numdam:PMIHES_1973__42__47_0 · eudml:103922 [19] P. Schenzel, On formal local cohomology and connectedness , J. Alg. 315 (2007), 897-923. · Zbl 1131.13018 · doi:10.1016/j.jalgebra.2007.06.015 [20] —-, On the use of local cohomology in algebra and geometry , in Six lectures in commutative algebra , J. Elias, J.M. Giral, R.M. Miró-Roig and S. Zarzuela, eds., Progr. Math. 166 , Birkhäuser, Berlin, 1998. · doi:10.1007/978-3-0346-0329-4_4 [21] —-, Proregular sequences, local cohomology, and completion , Math. Scand. 92 (2003), 161-180. · Zbl 1023.13011 · doi:10.7146/math.scand.a-14399 [22] A. Tehranian and A.P.E Talemi, Non-Artinian local cohomology with respect to a pair of ideals , A Colloquium 20 (2013), 637-642. · Zbl 1282.13038 · doi:10.1142/S1005386713000606 [23] T. Takahashi, Y. Yoshino and T. Yoshizawa, Local cohomology based on a nonclosed support defined by a pair of ideals , J. Pure Appl. Alg. 213 (2009), 582-600. · Zbl 1160.13013 · doi:10.1016/j.jpaa.2008.09.008 · arxiv:0709.3149 [24] C.A. Weibel, An introduction to homological algebra , Cambridge University Press, Cambridge, 1994.
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