Equimultiple coefficient ideals

  • P. H. Lima
  • V. H. Jorge Pérez

Abstract

Let $(R,\mathfrak {m})$ be a quasi-unmixed local ring and $I$ an equimultiple ideal of $R$ of analytic spread $s$. In this paper, we introduce the equimultiple coefficient ideals. Fix $k\in \{1,\dots ,s\}$. The largest ideal $L$ containing $I$ such that $e_{i}(I_{\mathfrak{p} })=e_{i}(L_{\mathfrak{p} })$ for each $i \in \{1,\dots ,k\}$ and each minimal prime $\mathfrak{p} $ of $I$ is called the $k$-th equimultiple coefficient ideal denoted by $I_{k}$. It is a generalization of the coefficient ideals introduced by Shah for the case of $\mathfrak {m}$-primary ideals. We also see applications of these ideals. For instance, we show that the associated graded ring $G_{I}(R)$ satisfies the $S_{1}$ condition if and only if $I^{n}=(I^{n})_{1}$ for all $n$.

References

Brumatti, P., Simis, A., and Vasconcelos, W. V., Normal Rees algebras, J. Algebra 112 (1988), no. 1, 26–48. https://doi.org/10.1016/0021-8693(88)90130-5

Ciupercă, C., First coefficient ideals and the $rm S_2$-ification of a Rees algebra, J. Algebra 242 (2001), no. 2, 782–794. https://doi.org/10.1006/jabr.2001.8835

Corso, A. and Polini, C., Links of prime ideals and their Rees algebras, J. Algebra 178 (1995), no. 1, 224–238. https://doi.org/10.1006/jabr.1995.1346

Grothe, U., Herrmann, M., and Orbanz, U., Graded Cohen-Macaulay rings associated to equimultiple ideals, Math. Z. 186 (1984), no. 4, 531–556. https://doi.org/10.1007/BF01162779

Heinzer, W., Johnston, B., Lantz, D., and Shah, K., Coefficient ideals in and blowups of a commutative Noetherian domain, J. Algebra 162 (1993), no. 2, 355–391. https://doi.org/10.1006/jabr.1993.1261

Heinzer, W., Lantz, D., and Shah, K., The Ratliff-Rush ideals in a Noetherian ring, Comm. Algebra 20 (1992), no. 2, 591–622. https://doi.org/10.1080/00927879208824359

Herrmann, M., Ikeda, S., and Orbanz, U., Equimultiplicity and blowing up: an algebraic study, Springer-Verlag, Berlin, 1988. https://doi.org/10.1007/978-3-642-61349-4

Noh, S. and Vasconcelos, W. V., The $S_2$-closure of a Rees algebra, Results Math. 23 (1993), no. 1-2, 149–162. https://doi.org/10.1007/BF03323133

Puthenpurakal, T. J. and Zulfeqarr, F., Ratliff-Rush filtrations associated with ideals and modules over a Noetherian ring, J. Algebra 311 (2007), no. 2, 551–583. https://doi.org/10.1016/j.jalgebra.2007.01.006

Shah, K., Coefficient ideals, Trans. Amer. Math. Soc. 327 (1991), no. 1, 373–384. https://doi.org/10.2307/2001847

>

Published
2017-09-22
How to Cite
Lima, P., & Jorge Pérez, V. (2017). Equimultiple coefficient ideals. MATHEMATICA SCANDINAVICA, 121(1), 5-18. https://doi.org/10.7146/math.scand.a-25988
Section
Articles