Hostname: page-component-cd9895bd7-gxg78 Total loading time: 0 Render date: 2024-12-25T06:33:40.589Z Has data issue: false hasContentIssue false
  • English
  • Français

NONINCREASING DEPTH FUNCTIONS OF MONOMIAL IDEALS

Part of: General commutative ring theory Theory of modules and ideals

Published online by Cambridge University Press:  28 January 2018

KAZUNORI MATSUDA ,
TAO SUZUKI  and
AKIYOSHI TSUCHIYA
KAZUNORI MATSUDA
Affiliation:
Department of Pure and Applied Mathematics, Graduate School of Information Science and Technology, Osaka University, Suita, Osaka 565-0871, Japan e-mail: kaz-matsuda@ist.osaka-u.ac.jp, t-suzuki@ist.osaka-u.ac.jp, a-tsuchiya@ist.osaka-u.ac.jp
TAO SUZUKI
Affiliation:
Department of Pure and Applied Mathematics, Graduate School of Information Science and Technology, Osaka University, Suita, Osaka 565-0871, Japan e-mail: kaz-matsuda@ist.osaka-u.ac.jp, t-suzuki@ist.osaka-u.ac.jp, a-tsuchiya@ist.osaka-u.ac.jp
AKIYOSHI TSUCHIYA
Affiliation:
Department of Pure and Applied Mathematics, Graduate School of Information Science and Technology, Osaka University, Suita, Osaka 565-0871, Japan e-mail: kaz-matsuda@ist.osaka-u.ac.jp, t-suzuki@ist.osaka-u.ac.jp, a-tsuchiya@ist.osaka-u.ac.jp
Rights & Permissions [Opens in a new window]

Abstract

Core share and HTML view are not available for this content. However, as you have access to this content, a full PDF is available via the ‘Save PDF’ action button.

Given a nonincreasing function f : ℤ≥ 0 \{0} → ℤ≥ 0 such that (i) f(k) − f(k + 1) ≤ 1 for all k ≥ 1 and (ii) if a = f(1) and b = limk → ∞f(k), then |f−1(a)| ≤ |f−1(a − 1)| ≤ ··· ≤ |f−1(b + 1)|, a system of generators of a monomial ideal I ⊂ K[x1, . . ., xn] for which depth S/Ik = f(k) for all k ≥ 1 is explicitly described. Furthermore, we give a characterization of triplets of integers (n, d, r) with n > 0, d ≥ 0 and r > 0 with the properties that there exists a monomial ideal I ⊂ S = K[x1, . . ., xn] for which limk→∞ depth S/Ik = d and dstab(I) = r, where dstab(I) is the smallest integer k0 ≥ 1 with depth S/Ik0 = depth S/Ik0+1 = depth S/Ik0+2 = ···.

MSC classification

Primary: 13A02: Graded rings
Secondary: 13A15: Ideals; multiplicative ideal theory 13C15: Dimension theory, depth, related rings (catenary, etc.)
Type
Research Article
Information
Glasgow Mathematical Journal , Volume 60 , Issue 2 , May 2018 , pp. 505 - 511
Copyright
Copyright © Glasgow Mathematical Journal Trust 2018 

References

REFERENCES

1. Brodmann, M., The asymptotic nature of the analytic spread, Math. Proc. Cambridge Philos. Soc. 86 (1979), 35–39.CrossRefGoogle Scholar
2. Hà, H. T., Nguyen, H. D., Trung, N. V. and Trung, T. N., Symbolic powers of sums of ideals, arXiv:1702.01766.Google Scholar
3. Hà, H. T., Trung, N. V. and Trung, T. N., Depth and regularity of powers of sums of ideals, Math. Z. 282 (2016), 819–838.CrossRefGoogle Scholar
4. Herzog, J. and Hibi, T., The depth of powers of an ideal, J. Algebra 291 (2005), 534–550.CrossRefGoogle Scholar
5. Miller, E. and Sturmfels, B., Combinatorial commutative algebra, Graduate Texts in Mathematics, Vol. 227 (Springer-Verlag, New York, 2005).Google Scholar
6. Nguyen, H. D. and Vu, T., Powers of sums and their homological invariants, arXiv:1607.07380.Google Scholar