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Differential operators, retracts, and toric face rings

Christine Berkesch, C-Y. Jean Chan, Patricia Klein, Laura Felicia Matusevich, Janet Page and Janet Vassilev

Vol. 17 (2023), No. 11, 1959–1984
Abstract

We give explicit descriptions of rings of differential operators of toric face rings in characteristic 0. For quotients of normal affine semigroup rings by radical monomial ideals, we also identify which of their differential operators are induced by differential operators on the ambient ring. Lastly, we provide a criterion for the Gorenstein property of a normal affine semigroup ring in terms of its differential operators.

Our main technique is to realize the k-algebras we study in terms of a suitable family of their algebra retracts in a way that is compatible with the characterization of differential operators. This strategy allows us to describe differential operators of any k-algebra realized by retracts in terms of the differential operators on these retracts, without restriction on char (k).

Keywords
differential operators, toric face rings, algebra retracts, affine semigroup rings
Mathematical Subject Classification
Primary: 16S32
Secondary: 13F55, 13N05
Milestones
Received: 31 January 2022
Revised: 18 January 2023
Accepted: 23 February 2023
Published: 3 October 2023
Authors
Christine Berkesch
School of Mathematics
University of Minnesota
Minneapolis
MN
United States
C-Y. Jean Chan
Department of Mathematics
Central Michigan University
Mount Pleasant
MI
United States
Patricia Klein
Department of Mathematics
Texas A&M University
College Station
TX
United States
Laura Felicia Matusevich
Department of Mathematics
Texas A&M University
College Station
TX
United States
Janet Page
Mathematics Department
North Dakota State University
Fargo
ND
United States
Janet Vassilev
Department of Mathematics and Statistics
University of New Mexico
Albuquerque
NM
United States

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