Remote Access Proceedings of the American Mathematical Society
Green Open Access

Proceedings of the American Mathematical Society

ISSN 1088-6826(online) ISSN 0002-9939(print)

 
 

 

A linear function associated to asymptotic prime divisors


Authors: Daniel Katz and Eric West
Journal: Proc. Amer. Math. Soc. 132 (2004), 1589-1597
MSC (2000): Primary 13A02, 13A15, 13A30, 13E05
DOI: https://doi.org/10.1090/S0002-9939-03-07282-4
Published electronically: October 21, 2003
MathSciNet review: 2051118
Full-text PDF

Abstract | References | Similar Articles | Additional Information

Abstract: Let $R$ be a Noetherian standard ${\mathbb{N}}^{\thinspace d}$-graded ring and $M,N$ finitely generated, ${\mathbb{N}}^{\thinspace d}$-graded $R$-modules. Let $I_{1}, \ldots , I_{s}$ be finitely many homogeneous ideals of $R$. We show that there exist linear functions $f,g : \mathbb{N}^{s} \to \mathbb{N}^{d}$such that the associated primes over $R_{0}$ of $[\operatorname{Ext}^{i}(N,M/I_{1}^{n_{1}}\cdots I_{s}^{n_{s}}M)]_{m}$ and $[\operatorname{Tor}_{i}(N,M/I_{1}^{n_{1}}\cdots I_{s}^{n_{s}}M)]_{m}$ are stable whenever $m\in {\mathbb{N}}^{\thinspace d}$ satisfies $m\geq f(n_{1},\ldots ,n_{s})$ and $m\geq g(n_{1},\ldots , n_{s})$, respectively.


References [Enhancements On Off] (What's this?)

  • [B] M. Brodmann, Asymptotic stability of $Ass (M/I^{n}M)$, Proc. Amer. Math. Soc. 74 (1979), 16-18. MR 80c:13012
  • [CHT] S. Cutkosky, J. Herzog, and N. Trung, Asymptotic behaviour of Castelnuovo-Mumford regularity, Compositio Math. 118 (1999), 243-261. MR 2000f:13037
  • [KMR] D. Katz, S. McAdam and L. J. Ratliff, Jr., Prime divisors and divisorial ideals, J. Pure Appl. Algebra 59 (1989), 179-186. MR 90g:13030
  • [Ko] V. Kodiyalam, Asymptotic behaviour of Castelnuovo-Mumford regularity, Proc. Amer. Math. Soc. 128 (2000), 407-411. MR 2000c:13027
  • [KS] A. K. Kingsbury and R. Y. Sharp, Asymptotic behaviour of certain sets of prime ideals, Proc. Amer. Math. Soc. 124 (1996), 1703-1711. MR 96h:13003
  • [Mc] S. McAdam, Asymptotic Prime Divisors, Lecture Notes in Math., vol. 1023, Springer-Verlag, New York, 1983. MR 85f:13018
  • [Sh] R. Y. Sharp, Injective modules and linear growth of primary decompositions, Proc. Amer. Math. Soc. 128 (2000), 717-722. MR 2000e:13004
  • [Si] A. Singh, $p$-torsion elements in local cohomology modules, Math. Res. Lett. 7 (2000), 165-176. MR 2001g:13039
  • [Sw] I. Swanson, Powers of ideals. Primary decomposition, Artin-Rees lemma and regularity, Math. Ann. 307 (1997), 299-313. MR 97j:13005
  • [T] E. Theodorescu, Derived functors and Hilbert polynomials, Math. Proc. Cambridge Philos. Soc. 132 (2002), 75-88. MR 2002j:13018
  • [W] E. West, Primes associated to multigraded modules, J. Algebra (to appear).
  • [Y] Y. Yao, Ph. D. Thesis, University of Kansas (2002).

Similar Articles

Retrieve articles in Proceedings of the American Mathematical Society with MSC (2000): 13A02, 13A15, 13A30, 13E05

Retrieve articles in all journals with MSC (2000): 13A02, 13A15, 13A30, 13E05


Additional Information

Daniel Katz
Affiliation: Department of Mathematics, University of Kansas, Lawrence, Kansas 66045
Email: dlk@math.ukans.edu

Eric West
Affiliation: Department of Mathematics and Computer Science, Benedictine College, Atchison, Kansas 66002
Email: ewest@benedictine.edu

DOI: https://doi.org/10.1090/S0002-9939-03-07282-4
Keywords: Associated prime, multi-graded module, homology module
Received by editor(s): April 8, 2002
Received by editor(s) in revised form: February 13, 2003
Published electronically: October 21, 2003
Communicated by: Bernd Ulrich
Article copyright: © Copyright 2003 American Mathematical Society

American Mathematical Society