On the Frobenius Complexity of Stanley-Reisner Rings

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 $-\mathrm{\infty }$ 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 $e\ge 2,$ generalizing work of Alvarez Montaner, Boix and Zarzuela. Our result settles an open question mentioned by Alvarez Montaner in one of his papers.