Vol. 14, No. 8, 2020

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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