نتایجی در مورد خاصیت آرتینی مدول های کوهمولوژی موضعی در نقطه ارتفاع یک ایده آل
Subject Areas : Algebra
MirYousef Sadeghi
1
,
Khadijeh Ahmadi Amoli
2
,
Maryam Chaghamirza
3
1 - Department of Mathematics, Payame Noor University, Tehran, Iran
2 - Department of Mathematics, Payame Noor University, Tehran, Iran
3 - Department of Mathematics, Payame Noor University, Tehran, Iran
Keywords: Local cohomology modules, Artinian modules, Local rings, Cohen-Macaulay rings, Analytically irreducible rings.,
Abstract :
Let R be a commutative Noetherian ring, a be a non-zero ideal of R, and M be a finitely generated R-module. We first show that, if IM≠M and MinAss_R (M/IM)⊆Min(I)\Max(R), then Supp_R H_I^(ht_M I) (M)⊈Max(R), and so the R-module H_I^(ht_M I) (M) is not Artinian. As a consequence, if Min(I)\Max(R)≠∅, then the R-module H_I^(htI) (R) is not Artinian, considering M:=R. Later, we give some results on Artinianness of the R-module H_I^(dimR-1) (R), when R is a local ring. Throughout this article, $R$ denotes a commutative Noetherian ring with non-zero identity, $\fa$ denotes a proper non-zero ideal of $R$, and $M$ denotes a finitely generated $R$-module. By $\mathbb{N}_{0}$, we mean the set of non-negative integers. Moreover, we use $\Min(\fa)$ (resp. $\Max(R)$) to denote the set of minimal prime ideals of $\fa$ (resp. the set of maximal ideals of $R$). The $i$-th local cohomology module of $M$ is defined by $$H_{\mathfrak{a}}^{i}(M):= \displaystyle\lim_ {\overrightarrow{n\in \mathbb{N}}}\mathrm{Ext}_{R}^{i} \left( {R}/{\mathfrak{a}^{n}},M\right).$$
[1] M. P. Brodmann and R. Y. Sharp, Local cohomology: an algebraic introduction with geometric applications. 2nd ed. Cambridge Studies in Advanced Mathematics. Cambridge University Press (2013).
[2] R. Y. Sharp, Steps in commutative Algebra. Second edition. London Mathematical Society Student Text 19, Cambridge University Press (2000).
[3] C. Huneke, Problems on local cohomology, Free Resolutions in commutative algebra and algebraic geometry (Sundance, Utah, 1990), Research Notes in Mathematics 2, Boston, Ma, Jones and Bartlett Publisher, (1992), 93-108.
[4] A. Ogus, Local cohomological dimension of algebraic varieties. Annals of Math., 98 (1973), 327-365.
[5] R. Hartshorne and R. Speiser, Local cohomological dimension in characteristic p. Ann. Math., 105 (1977), 45-79.
[6] T. Marley, The associated primes of local cohomology modules over rings of small dimension. manuscripta math., 104 (2001), 519-525.
[7] M. Aghapournahr and K. Bahmanpour, Cofiniteness of weakly Laskerian local cohomology modules. Bull. Math. Soc. Math. Romanie, 105 (2014), 347-365.
[8] D. Asadollahi and R. Naghipour, Faltings' Local-global Principle for the Finiteness of local cohomology modules. Commun. Algebra, 43 (2015), 953-958.
[9] L. Melkersson, Modules cofinite whit respect to an ideal. J. Algebra, 285 (2005), 649-668.
[10] R. Hartshorne, Affine duality and cofiniteness. Invent. Math., 9 (1970), 145-164.
[11] K. I. Yoshida, Cofiniteness of local cohomology modules for ideals of dimension one. Nagoya Math. J., 147 (1997), 179-191.