On the Frobenius complexity of Stanley–Reisner rings

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 −∞ or 0. This invariant is determined by the complexity sequence ${\left\{c_e\right\}}_e$ of the ring of Frobenius operators on the injective hull of the residue field. We will show that ${\left\{c_e\right\}}_e$ is constant for $e\ge 2,$ generalizing work of Àlvarez Montaner, Boix, and Zarzuela. Our result settles an open question mentioned by Àlvarez Montaner.

Keywords:

• Commutative ring
• complexity sequence
• Frobenius algebra
• Frobenius operators
• Stanley–Reisner ring

MATHEMATICS SUBJECT CLASSIFICATION:

• Primary: 13A35
• Secondary: 13F55

Acknowledgments

I would like to thank my advisor, Florian Enescu, for his constant support and guidance and Yongwei Yao for useful comments.