The core of monomial ideals

Fouli, Louiza; Montaño, Jonathan; Polini, Claudia; Ulrich, Bernd

doi:10.2140/ant.2025.19.1463

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The core of monomial ideals

Louiza Fouli, Jonathan Montaño, Claudia Polini and Bernd Ulrich

Vol. 19 (2025), No. 8, 1463–1494

DOI: 10.2140/ant.2025.19.1463

Abstract

The core of an ideal is defined as the intersection of all of its reductions. We provide an explicit description for the core of a monomial ideal $I$ satisfying certain residual conditions, showing that $core (I)$ coincides with the largest monomial ideal contained in a general reduction of $I$ . We prove that the class of lex-segment ideals satisfies these residual conditions and study the core of lex-segment ideals generated in one degree. For monomial ideals that do not necessarily satisfy the residual conditions and that are generated in one degree, we conjecture an explicit formula for the core, and make progress towards this conjecture.

Keywords

monomial ideals, reductions, core, lex-segment ideals, canonical module, special fiber rings, adjoints

Mathematical Subject Classification

Primary: 05E40, 13A30, 13B22

Secondary: 13C40, 13P10, 14F18

Milestones

Received: 27 May 2023

Revised: 22 November 2023

Accepted: 13 February 2024

Published: 12 June 2025

Authors

Louiza Fouli
Department of Mathematical Sciences New Mexico State University Las Cruces, NM United States

Jonathan Montaño
Arizona State University Tempe, AZ United States

Claudia Polini
Department of Mathematics University of Notre Dame Notre Dame, IN United States

Bernd Ulrich
Department of Mathematics Purdue University West Lafayette, IN United States

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Vol. 19, No. 8, 2025