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Stability of the depth function of good filtrations | ||
| Journal of Algebra and Related Topics | ||
| دوره 13، شماره 2، اسفند 2025، صفحه 199-208 اصل مقاله (173.3 K) | ||
| نوع مقاله: Research Paper | ||
| شناسه دیجیتال (DOI): 10.22124/jart.2025.27064.1654 | ||
| نویسندگان | ||
| D. F. J. Kouakou* 1؛ A. Assane1؛ D. Kamano2 | ||
| 1Laboratoire de Mathématiques et Informatique, Université Nangui Abrogoua, UFR - SFA, Côte d'Ivoire | ||
| 2Departement de Sciences et Technologie, Ecole Normale Supérieure d'Abidjan, Côte d'Ivoire | ||
| چکیده | ||
| Let $A$ be a Noetherian local ring, and let $I \subset J$ be two ideals of $A$. Let $M$ be a finitely generated $A$-module. Brodmann proved that the function $n \mapsto \mathrm{depth}_{J}(\frac{M}{I^{n}M})$ is constant for large $n$. In this paper, we consider a filtration $\phi = (M_n)_{n \in \mathbb{N}}$ of $M$ and a filtration $f = (I_n)_{n \in \mathbb{N}}$ of $A$. Generalizing Brodmann's result, we first show that the function $n \mapsto \mathrm{depth}_{J}(\frac{M}{M_{n}})$ is constant for large $n$ of value $\mathrm{depth}_{J}(f, M)$, provided that $\phi$ is $f$-good and $f$ is strongly Noetherian. Secondly, we establish the inequality $\gamma_{J}(f, M) \leq \dim_A(M) - \mathrm{depth}_{J}(f, M)$, where $\gamma_{J}(f, M)$ denotes the analytic spread of $f$ at $J$ with respect to $M$. | ||
| کلیدواژهها | ||
| Filtration؛ dimension؛ analytic spread؛ depth | ||
| مراجع | ||
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[1] S. Bandari, J. Herzog and T. Hibi, Monomial ideals whose depth function has any number of strict local maxima, Arkiv för Matematik, (1) 52 (2014) 11–19. [2] W. Bishop, J. W. Petro, L. J. Ratliff and D. E. Rush, Note on Noetherian filtrations, Communications in Algebra, (2) 17 (1989) 471–485. [3] M. Brodmann, The asymptotic nature of the analytic spread, Mathematical Proceedings of the Cambridge Philosophical Society, (1) 86 (1979) 35–39. [4] M. Brodmann, Asymptotic stability of Ass( MInM ), Proceedings of the American Mathematical Society, (1) 74 (1979) 16–18. [5] W. Bruns and J. Herzog, Cohen-Macaulay Rings, Cambridge University Press (1993). [6] L. Burch, Codimension and analytic spread, Mathematical Proceedings of the Cambridge Philosophical Society, (3) 72 (1972) 369–373. [7] Y. Diagana, H. Dichi and D. Sangaré, Filtrations, generalized analytic independence, analytic spread, Afrika Mathematika, 4 (1994) 101–114. [8] D. Kamano, K. A. Essan, A. Abdoulaye and E. D. Akeke, σ-Sporadic prime ideals and superficial elements, Journal of Algebra and Related Topics, (2) 5 (2017) 35-45. [9] H. Matsumura, Commutative ring theory, Cambridge University Press (1989). [10] L. D. Nam and M. Varbaro, When does depth stabilize early on?, Journal of Algebra, 445 (2016) 181–192. | ||
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