پیوندهای مفید
آمار
| تعداد نشریات | 21 |
| تعداد شماره‌ها | 671 |
| تعداد مقالات | 9,762 |
| تعداد مشاهده مقاله | 69,474,475 |
| تعداد دریافت فایل اصل مقاله | 48,617,843 |
Some results on the vanishing and coassociated prime ideals of the top generalized local cohomology module | ||
| International Journal of Nonlinear Analysis and Applications | ||
| مقالات آماده انتشار، اصلاح شده برای چاپ، انتشار آنلاین از تاریخ 17 مهر 1404 اصل مقاله (349.5 K) | ||
| نوع مقاله: Research Paper | ||
| شناسه دیجیتال (DOI): 10.22075/ijnaa.2025.34639.5185 | ||
| نویسنده | ||
| Ali Fathi* | ||
| Department of Mathematics, Zanjan Branch, Islamic Azad University, Zanjan, Iran | ||
| تاریخ دریافت: 14 تیر 1403، تاریخ پذیرش: 07 اسفند 1403 | ||
| چکیده | ||
| Let $R$ be a commutative Noetherian ring, and let $\mathfrak a$ be a proper ideal of $R$. Let $M$ be a non-zero finitely generated $R$-module with the finite projective dimension $p$, and let $N$ be a non-zero finitely generated $R$-module of dimension $d$. Assume that $c$ is the greatest non-negative integer with the property that $\operatorname{H}^i_{\mathfrak a}(M, N)$, the $i$-th generalized local cohomology module of $M, N$ with respect to $\mathfrak a$, is non-zero. It is known that $\operatorname{H}^i_{\mathfrak a}(M, N)$ is zero for all $i>p+d$ and the top generalized local cohomology $\operatorname{H}^{p+d}_{\mathfrak a}(M, N)$ is Artinian. In this paper, we study the vanishing and attached primes of $\operatorname{H}^{p+d}_{\mathfrak a}(M, N)$. Also, we prove that if $\operatorname{H}^{p+r}_{\mathfrak a}(M, R/\mathfrak p)=0$ for a fixed non-negative integer $r$ and for all $\mathfrak p$ in $\operatorname {supp}_R(N)$, then $\operatorname{H}^{p+s}_{\mathfrak a}(M, N)=0$ for all $s\geq r$ and so $c<p+r$. We deduce that if $p\leq c$, then $$c=p+\min\{t\in\mathbb N_0: \operatorname{H}^{p+t}_{\mathfrak a}(M, R/\mathfrak p)=0 \text{ for all } \mathfrak p\in\operatorname {supp}_R(N)\}-1.$$ Also, we prove that, for each $i$ with $p\leq i\leq c$, there exists $\mathfrak p_i$ in $\operatorname{supp}_R(N)$ such that $\operatorname{H}^{i}_{\mathfrak a}(M, R/\mathfrak p_i)\neq 0$. | ||
| کلیدواژه‌ها | ||
| Generalized local cohomology module؛ cohomological dimension؛ coassociated prime ideal؛ attached prime ideal | ||
| مراجع | ||
|
[1] J. Amjadi and R. Naghipour, Cohomological dimension of generalized local cohomology modules, Algebra Colloq. 15 (2008), no. 2, 303–308. [2] M.H. Bijan-Zadeh, A common generalization of local cohomology theories, Glasgow Math. J. 21 (1980), no. 2, 173–181. [3] M.P. Brodmann and R.Y. Sharp, Local Cohomology: An Algebraic Introduction with Geometric Applications, Cambridge Studies in Advanced Mathematics 60, Cambridge University Press, Cambridge, 1998. [4] M.T. Dibaei and S. Yassemi, Attached primes of the top local cohomology modules with respect to an ideal, Arch. Math. (Basel) 84 (2005), no. 4, 292–297. [5] M.T. Dibaei and S. Yassemi, Top local cohomology modules, Algebra Colloq. 14 (2007), no. 2, 209–214. [6] K. Divaani-Aazar, Vanishing of the top local cohomology modules over Noetherian rings, Proc. Indian Acad. Sci. Math. Sci. 119 (2009), no. 1, 23–35. [7] K. Divaani-Aazar and A. Hajikarimi, Generalized local cohomology modules and homological Gorenstein dimensions, Comm. Algebra 39 (2011), no. 6, 2051–2067. [8] A. Fathi, The first non-isomorphic local cohomology modules with respect to their ideals, J. Algebra Appl. 17 (2018), no. 12, 1850230, 12. [9] A. Fathi, Lichtenbaum-Hartshorne vanishing theorem for generalized local cohomology modules, J. Adv. Math. Model. 13 (2023), no. 2, 250–258. [10] A. Fathi, A. Tehranian, and H. Zakeri, Filter regular sequences and generalized local cohomology modules, Bull. Malays. Math. Sci. Soc. 38 (2015), no. 2, 467–482. [11] Y. Gu and L. Chu, Attached primes of the top generalized local cohomology modules, Bull. Aust. Math. Soc. 79 (2009), no. 1, 59–67. [12] R. Hartshorne, Cohomological dimension of algebraic varieties, Ann. of Math. (2) 88 (1968), 403–450. [13] S.H. Hassanzadeh and A. Vahidi, On vanishing and cofiniteness of generalized local cohomology modules, Comm. Algebra 37 (2009), no. 7, 2290–2299. [14] J. Herzog, Komplexe, Auflösungen und Dualität in der Localen Algebra, Habilitationsschrift, Universität Regensburg, 1970. [15] I.G. Macdonald, Secondary representation of modules over a commutative ring, Symp. Math. 11 (1973), 23–43. [16] A. Mafi, On the associated primes of generalized local cohomology modules, Comm. Algebra 34 (2006), no. 7, 2489–2494. [17] S. Yassemi, Generalized section functors, J. Pure Appl. Algebra 95 (1994), no. 1, 103–119. [18] S. Yassemi, Coassociated primes, Comm. Algebra 23 (1995), no. 4, 1473–1498. [19] S. Yassemi, L. Khatami, and T. Sharif, Associated primes of generalized local cohomology modules, Comm. Algebra 30 (2002), no. 1, 327–330. | ||
|
آمار تعداد مشاهده مقاله: 121 تعداد دریافت فایل اصل مقاله: 55 |
||