Depth and regularity modulo a principal ideal
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We study the relationship between depth and regularity of a homogeneous ideal I and those of (I, f) and I : f, where f is a linear form or a monomial. Our results have several interesting consequences on depth and regularity of edge ideals of hypergraphs and of powers of ideals.
KeywordsDepth Regularity Monomial ideal Powers of an ideal Edge ideal Very well-covered graph Chordal graph
Mathematics Subject ClassificationPrimary 13C20 Secondary 13D45 14B05 05C65
Giulio Caviglia is partially supported by the Simons Foundation (Grant #209661). Huy Tài Hà is partially supported by the Simons Foundation (Grant #279786). Ngo Viet Trung is partially supported by Vietnam National Foundation for Science and Technology Development (Grant #101.04-2017.19) and the project VAST.HTQT.NHAT.1/16-18. The main part of this work was done during research stays of the authors at the American Institute of Mathematics in the SQuaRE program “Ordinary powers and symbolic powers” during the period 2012–2014. The authors are grateful to S. A. Seyed Fakhari for pointing out a mistake of Theorem 5.1 in the first version of this paper and to an anonymous referee for mentioning that Lemma 4.1(i) and (ii) were already proved in  and , respectively.
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