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Some applications of $k$-regular sequences and arithmetic rank of an ideal with respect to modules | ||
| Journal of Algebra and Related Topics | ||
| دوره 11، شماره 2، اسفند 2023، صفحه 21-35 اصل مقاله (133.69 K) | ||
| نوع مقاله: Research Paper | ||
| شناسه دیجیتال (DOI): 10.22124/jart.2023.24393.1518 | ||
| نویسندگان | ||
| Kh. Ahmadi Amoli* ؛ Z. Habibi؛ R. Behboodi | ||
| Department of Mathematics, Payame Noor University, Tehran, Iran | ||
| چکیده | ||
| Let $R$ be a commutative Noetherian ring with identity, $I$ be an ideal of $R$, and $M$ be an $R$-module. Let $k\geqslant -1$ be an arbitrary integer. In this paper, we introduce the notions of $\Rad_M(I)$ and $\ara_M(I)$ as the radical and the arithmetic rank of $I$ with respect to $M$, respectively. We show that the existence of some sort of regular sequences can be depended on $\dim M/IM$ and so, we can get some information about local cohomology modules as well. Indeed, if $\ara_M(I)=n\geq 1$ and ${(\Supp_{R}(M/IM))}_{>k}=\emptyset$ (e.g., if $\dim M/IM=k$), then there exist $n$ elements $x_1, ..., x_n$ in $I$ which is a poor $k$-regular $M$-sequence and generate an ideal with the same radical as $\Rad_M(I)$ and so $H^i_I(M)\cong H^i_{(x_1, ..., x_n)}(M)$ for all $i\in \mathbb{N}_0$. As an application, we show that $\ara_M(I) \leq \dim M+1$, which is a refinement of the inequality $\ara_R(I) \leq \dim R+1$ for modules, attributed to Kronecker and Forster. Then, we prove $\dim M-\dim M/IM \leq \cd(I, M) \leq \ara_M(I) \leq \dim M$, if $(R, \mathfrak{m})$ is a local ring and $IM \neq M$. | ||
| کلیدواژه‌ها | ||
| regular sequences؛ $k$-regular sequences؛ local cohomology modules؛ arithmetic rank of an ideal with respect to modules | ||
| مراجع | ||
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1. Kh. Ahmadi Amoli, Filter regular sequences, local cohomology modules and singular sets, Ph. D. Thesis, University for Teacher Education, Iran, 1996. 2. Kh. Ahmadi Amoli and N. Sanaei, On the k-regular sequences and the generalization of f-modules, J. Korean Math. Soc., (5) 49 (2012), 1083-1096. 3. J. Azami, Filter regular sequences and local cohomology modules, Journal of Algebraic System, (2) 7 (2020), 281-290. 4. K. Bahmanpour and R. Naghipour, Co niteness of local cohomology modules for ideals of small dimension, J. Algebra, 321 (2009), 1997-2011. 5. M. Brodmann and R. Y. Sharp, Local Cohomology; An Algebraic Introduction with Geometric Applications, Cambridge University Press, Cambridge, 1998. 6. N. Q. Chinh and L. T. Nhan, On the associated primes and the support of local cohomology modules, Algebra Colloq., (4) 15 (2008), 599-608. 7. O. Forster, Uber die Anzahl der Erzeugenden eines ideals in einem Noetherschen ring, (German) Math. Z., 84 (1964), 80-87. 8. A. Grothendieck, Local Cohomology, Notes by R. Hartshorne, Lecture Note in Math, Vol 864, Springer, New York, 1966. 9. D. Hassanzadeh-Lelekaami and H. Roshan-Shekalgourabi, On the laskerian properties of extension functors of local cohomology modules, J. Algebra Relat. Topics, (2) 10 (2022), 43-50. 10. M. Hellus and P. Schenzel, On cohomologically complete intersections, J. Algebra, (10) 320 (2008), 3733-3748. 11. L. Kronecker, Grundzug einer arithmetischen theorie der algebraischen Grossen, J. Reine Angew. Math., 92 (1882), 1-121. 12. G. Lyubeznik, A survey of problems and results on the number of de ning equations, Comm. Algebra, Springer, New York, (1989), 375-390. 13. G. Lyubeznik, The number of de ning equations of ane algebraic sets, Amer. J. Math., 114 (1992), 413-463. 14. H. Matsumura, Commutative Ring Theory, Cambridge University Press, Cambridge, 1986. 15. A. Mehrvarz, K. Bahmanpour, and R. Naghipour, Arithmetic rank, Cohomological dimension and lter regular sequences, J. Algebra Appl., (6) 8 (2009), 855-862. 16. L. Melkersson, Some applications of criterion for artinianness of a module, J. Pure Appl. Alg., (3) 101 (1995), 291-303. 17. L. T. Nhan, On generalized regular sequences and the niteness for associated primes of local cohomology modules, Comm. Algebra, (3) 33 (2005), 793-806. 18. P. Schenzel, On the Use of local Cohomology in Algebra and Geometry, Six lectures on commutative algebra (Bellaterra, 1996), 241-292. 19. P. Schenzel, N. V. Trung and N. T. Cuong, Verallgemeinerte Cohen-Macaulaymoduln , Math. Nachr., 85 (1978), 57-73. 20. J. -P. Serre, Faisceaux algebriques coherents, Annals of Math., 61 (1955), 197-278. 21. R. Y. Sharp, Steps in commutative algebra: second edition, London Mathematical Society student text 51, Cambridge University Press, Cambridge, 2000. | ||
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