Primary Ideals with Good Fiber Cone

Abstract

Let (R, 𝔪) be a Cohen–Macaulay local ring of dimension d > 0. I an 𝔪-primary ideal and K an ideal containing I. Let a ₁,…, a _d−1 ∈ I, a _d  ∈ K be a Rees-superficial sequence for I and K, we set J = (a ₁,…, a _d−1). In this article, we consider the classes of these 𝔪-primary ideals I such that KI ∩ (J, a _d ) = JK + a _d I and , or, for some positive integer k, KI ⁿ  ∩ (J, a _d ) = JKI ⁿ⁻¹ + a _d I ⁿ for n ≤ k − 1 and . We show that if depth G(I) ≥ d − 1 then depth F _K (I) ≥ d − 2. In these cases, we also compute the Hilbert series of F _K (I).

Key Words:

• Depth
• Fiber cone
• Hilbert series
• Joint reduction
• Joint reduction number
• Rees-superficial sequence

Mathematics Subject Classification:

• 13H10
• 13C15
• 13A15
• 13A30
• 13H15

ACKNOWLEDGMENTS

The author is grateful to Professor Zhong-Ming Tang for useful discussions. She would like to express her sincere thanks to the editor for help and encouragement. Special thanks are due to the referee for a careful reading and pertinent comments.

This research is partially supported by the National Natural Science Foundation of China (10771152), the Natural Science Foundation for Colleges and Universities in Jiangsu Province (No. 10KJB110007; 09KJB110006), and the Pre-research Project of Soochow University.

Notes

Communicated by I. Swanson.