Frobenius identities for the volume map on Cohen--Macaulay rings
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arXiv
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| Format: | Preprint |
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2026
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| author | Adiprasito, Karim Alexander Katz, Eric Oba, Ryoshun Papadakis, Stavros Argyrios Petrotou, Vasiliki |
| author_facet | Adiprasito, Karim Alexander Katz, Eric Oba, Ryoshun Papadakis, Stavros Argyrios Petrotou, Vasiliki |
| contents | 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2605_02479 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | Frobenius identities for the volume map on Cohen--Macaulay rings Adiprasito, Karim Alexander Katz, Eric Oba, Ryoshun Papadakis, Stavros Argyrios Petrotou, Vasiliki Commutative Algebra Combinatorics 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. |
| title | Frobenius identities for the volume map on Cohen--Macaulay rings |
| topic | Commutative Algebra Combinatorics |
| url | https://arxiv.org/abs/2605.02479 |