Frobenius identities for the volume map on Cohen--Macaulay rings

TL;DR

This paper explores the volume map on Artinian quotients of Cohen-Macaulay rings in characteristic p, linking it with Frobenius actions to understand identities and properties like Lefschetz and anisotropy.

Contribution

It introduces a conceptual framework connecting Frobenius actions and volume identities, leading to new proofs and conditions for algebraic properties in Cohen-Macaulay rings.

Findings

Established sufficient conditions for Lefschetz properties in Gorenstein rings.

Proved a Parseval-Rayleigh identity for codimension-3 Gorenstein quotients.

Deduced the g-theorem for simplicial spheres and confirmed the Ohsugi-Hibi conjecture.

Abstract

We study the volume map on Artinian quotients of Cohen-Macaulay algebras in characteristic $p$, and the interaction between it and the action of Frobenius on resolutions. This allows us to provide a general, conceptual way to understand Parseval-Rayleigh identities, curious inhomogeneous identities on the volume map which were developed for the proof of the Ohsugi-Hibi conjecture. This general perspective gives a new approach to generic Lefschetz theory. We use this perspective to do the following: we give sufficient conditions for anisotropy and the Hard Lefschetz property for generic Artinian reductions of graded Gorenstein rings; we study the codimension-$3$ Gorenstein quotient of a polynomial ring by the ideal generated by Pfaffians, proving a Parseval-Rayleigh identity and deriving anisotropy and Hard Lefschetz in characteristic $2$; we deduce the $g$-theorem for simplicial spheres and the Ohsugi-Hibi conjecture following previous work of Adiprasito, Papadakis, and Petrotou; and we provide further examples of Parseval-Rayleigh identities for Gorenstein rings.