A Lower Bound on the Second Fiber Coefficient of the Fiber Cones

Abstract

Let (R, 𝔪) be a Cohen–Macaulay local ring of dimension d > 0, I an 𝔪-primary ideal of R and K an ideal containing I. When depth G(I) ≥ d − 1 and r(I | K) < ∞, we present a lower bound on the second fiber coefficient of the fiber cones, and also provide a characterization, in terms of f ₂(I, K), of the condition depth F _K (I) ≥ d − 1.

Key Words:

• Depth
• Fiber coefficient
• Fiber cone
• Joint reduction
• Rees-superficial sequence

2000 Mathematics Subject Classification:

• 13C15
• 13A15
• 13A30
• 13H15

ACKNOWLEDGMENTS

We would like to express our sincere thanks to the editor for her help and encouragement. Special thanks are due to the referee for pointing out the correct assumptions in Theorem 3.6, a careful reading and some pertinent comments. This research was partially supported by the National Natural Science Foundation of China (10771152) and by Pre-research Project of Soochow University.

Notes

Communicated by I. Swanson.