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arXiv · 2605.02479

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

Karim Alexander Adiprasito·Eric Katz·Ryoshun Oba·Stavros Argyrios Papadakis·Vasiliki Petrotou
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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.

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Karim Alexander Adiprasito, Eric Katz, Ryoshun Oba, Stavros Argyrios Papadakis, Vasiliki Petrotou. 2026-05-04. Frobenius identities for the volume map on Cohen--Macaulay rings. https://arxiv.org/abs/2605.02479

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