Upper bound of multiplicity in Cohen-Macaulay rings of prime characteristic
Duong Thi Huong, Pham Hung Quy
TL;DR
This work bounds the multiplicity of a Cohen–Macaulay local ring of prime characteristic in terms of the Frobenius test exponent, dimension, embedding dimension, and type. Using tight closure theory, Briançon–Skoda, and Frobenius actions on local cohomology, it derives explicit parity-based inequalities with $Q=p^{\mathrm{Fte}(R)}$ that extend Huneke–Watanabe bounds from the Gorenstein case to all Cohen–Macaulay rings; an $F$-nilpotent refinement is provided. The results connect $\mathrm{Fte}(R)$ with $\mathrm{HSL}(R)$ and show how $F$-singularities constrain $e(R)$, offering sharper control over Hilbert–Samuel multiplicities in positive characteristic. The formulas specialize to the known bounds when $R$ is Gorenstein and align with prior results for $F$-pure and $F$-rational rings, contributing to a broader understanding of singularities via Frobenius invariants.
Abstract
Let $(R, \frak m)$ be a local ring of prime characteristic $p$ and of dimension $d$ with the embedding dimension $v$, type $s$ and the Frobenius test exponent for parameter ideals $\mathrm{Fte}(R)$. We will give an upper bound for the multiplicity of Cohen-Macaulay rings in prime characteristic in terms of $\mathrm{Fte}(R),d,v$ and $s$. Our result extends the main results for Gorenstein rings due to Huneke and Watanabe.