On the Comparison of Rees's Valuations Over a Quasi-Unmixed Local Ring

Abstract

In this paper, we will show that if (R, 𝔪) is a quasi-unmixed local ring, I an 𝔪-primary ideal of R and ℛ𝒱(I) is the set of Rees valuations of I, then the number of minimal prime ideals in the 𝔪-adic completion of R equals exactly the number of equivalence classes on the set ℛ𝒱(I) under the equivalence relation ∼defined by: ν₁ ∼ ν₂ if there exist a constant c ≥ 1 such that for all x ∈ R, ν₁(x) ≤ cν₂(x) and ν₂(x) ≤ cν₁(x).

Key Words:

• Divisorial valuations
• Quasi-unmixed local domains
• Rees algebra
• Rees valuations

2010 Mathematics Subject Classification:

• 13F30
• 13G05
• 13B35

ACKNOWLEDGMENTS

I would like to thank Professor Mark Spivakovsky for his useful comments that helped improve the clarity of this paper.

Notes

Communicated by I. Swanson.