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On coefficient ideals

  • R. CALLEJAS-BEDREGAL (a1), V. H. JORGE PÉREZ (a2) and M. DUARTE FERRARI (a3)

Abstract

Let (R, 𝔪) be a Noetherian local ring and I an arbitrary ideal of R with analytic spread s. In [3] the authors proved the existence of a chain of ideals II[s] ⊆ ⋅⋅⋅ ⊆ I[1] such that deg(PI[k]/I) < sk. In this article we obtain a structure theorem for this ideals which is similar to that of K. Shah in [10] for 𝔪-primary ideals.

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[1] Amao, J. O. On a certain Hilbert polynomial. J. London Math. Soc. (2) 14, 1 (1976), 1320.
[2] Callejas–Bedregal, R., Pérez, V. H. Jorge and Silva, M. D.. On coefficient modules and Ratliff–Rush closure. Preprint.
[3] Herzog, J., Puthenpurakal, T. and Verma, J. K. Hilbert polynomials and powers of ideals. Math. Proc. Camb. Phil. Soc. 145 (2008), 623642.
[4] Huneke, C. and Swanson, I. Integral closure of ideals, rings and modules. London Math. Soc. Lecture Note Series (Cambridge University Press, 2006).
[5] McAdam, S. Asymptotic prime divisors. Lecture Notes in Math. 1023 (Springer-Verlag, Berlin, 1983).
[6] Ratliff, L. J. and Rush, D. Two notes on reductions of ideals. Indiana University Math J. 27 (1978), 929934.
[7] Rees, D. $\Mathcal{A}$-transforms of local rings and a theorem on multiplicities of ideals. Cambridge Philos. Soc. 57 (1961), 817.
[8] Rees, D. Amao's theorem and reduction criteria. J. London Math. Soc. (2) 32 (1985), no. 3, 404410.
[9] Rees, D. Lectures on the asymptotic theory of ideals. London Math. Soc. Lecture Note Series, 113 (Cambridge University Press, Cambridge, 1988).
[10] Shah, K. K. Coefficient ideals Trans. Math. Soc. 327 (1991), 373384.
[11] Samuel, P. and Zariski, O. Commutative Algebra. Vol. II (Springer-Verlag, Berlin, 1975).

On coefficient ideals

  • R. CALLEJAS-BEDREGAL (a1), V. H. JORGE PÉREZ (a2) and M. DUARTE FERRARI (a3)

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