Burch ideals and Burch rings

Dao, Hailong; Kobayashi, Toshinori; Takahashi, Ryo

doi:10.2140/ant.2020.14.2121

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Burch ideals and Burch rings

Hailong Dao, Toshinori Kobayashi and Ryo Takahashi

Vol. 14 (2020), No. 8, 2121–2150

DOI: 10.2140/ant.2020.14.2121

Abstract

We introduce the notion of Burch ideals and Burch rings. They are easy to define, and can be viewed as generalization of many well-known concepts, for example integrally closed ideals of finite colength and Cohen–Macaulay rings of minimal multiplicity. We give several characterizations of these objects. We show that they satisfy many interesting and desirable properties: ideal-theoretic, homological, categorical. We relate them to other classes of ideals and rings in the literature.

Dedicated to Lindsay Burch

Keywords

Burch ideal, Burch ring, direct summand, fiber product, Gorenstein ring, hypersurface, singular locus, singularity category, syzygy, thick subcategory, (weakly) m-full ideal

Mathematical Subject Classification 2010

Primary: 13C13

Secondary: 13D09, 13H10

Milestones

Received: 12 June 2019

Revised: 23 November 2019

Accepted: 5 March 2020

Published: 18 September 2020

Authors

Hailong Dao
Department of Mathematics University of Kansas Lawrence, KS United States

Toshinori Kobayashi
Graduate School of Mathematics Nagoya University Nagoya Japan

Ryo Takahashi
Graduate School of Mathematics Nagoya University Nagoya Japan	Department of Mathematics University of Kansas Lawrence KS United States

Vol. 14, No. 8, 2020