On Frobenius Betti numbers of graded rings of finite Cohen-Macaulay type

TL;DR

This work computes Frobenius Betti numbers for a class of Cohen–Macaulay graded rings of finite CM type in prime characteristic, linking these invariants to the Hilbert–Kunz multiplicity and $F$-signature. It relies on the classification of such rings in prime characteristic $p>2$ (scrolls and Veronese-type invariant rings) and analyzes each non-regular, non-hypersurface case by decomposing Frobenius powers into indecomposable maximal Cohen–Macaulay modules. The paper provides explicit closed-form formulas for $s(R)$, $e_{HK}(R)$, and $\beta_i^F(R)$ for three standard CM-type families: $R=\bk[x^\delta,\dots,y^\delta]$, $R=\bk[x^2,xy,y^2,xz,yz]$, and $R=\bk[x^2,y^2,z^2,xy,xz,yz]$ (with $\operatorname{char}\bk\neq2$). These results yield concrete benchmarks for singularity measures in finite CM type rings and illustrate how Frobenius asymptotics can be computed via decompositions into indecomposable MCM modules.

Abstract

The notion of Frobenius Betti numbers generalizes the Hilbert-Kunz multiplicity theory and serves as an invariant that measures singularity. However, the explicit computation of the Frobenius Betti numbers of rings has been limited to very few specific cases. This article focuses on the explicit computation of Frobenius Betti numbers of Cohen-Macaulay graded rings of finite Cohen-Macaulay type.

Theorems & Definitions (15)

• Proposition 1.1
• Proposition 2.2
• Theorem 2.3
• Lemma 4.2
• proof
• Theorem 4.5
• proof
• Lemma 5.5
• proof
• Theorem 5.10