On the Frobenius Complexity of Stanley-Reisner Rings

TL;DR

This paper investigates the Frobenius complexity of Stanley-Reisner rings, showing that the complexity sequence stabilizes for large e, thereby resolving an open question and deepening understanding of Frobenius operators in these rings.

Contribution

It proves that the Frobenius complexity sequence is constant for e ≥ 2 in Stanley-Reisner rings, generalizing previous work and settling an open problem.

Findings

The Frobenius complexity of Stanley-Reisner rings is either -∞ or 0.

The complexity sequence {c_e} is constant for e ≥ 2.

The result confirms a conjecture and extends prior research.

Abstract

The Frobenius complexity of a local ring $R$ measures asymptotically the abundance of Frobenius operators of order $e$ on the injective hull of the residue field of $R$. It is known that, for Stanley-Reisner rings, the Frobenius complexity is either $-\infty$ or $0$. This invariant is determined by the complexity sequence $\{c_ e\}_e $ of the ring of Frobenius operators on the injective hull of the residue field. We will show that $\{c_ e\}_e $ is constant for $e\geq 2,$ generalizing work of Alvarez Montaner, Boix and Zarzuela. Our result settles an open question mentioned by Alvarez Montaner in one of his papers.