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Generalizing Fröberg's Theorem on Ideals with Linear Resolutions

In 1990, Fröberg presented a combinatorial classification of the quadratic square-free monomial ideals with linear resolutions. He showed that the edge ideal of a graph has a linear resolution if and only if the complement of the graph is chordal. Since then, a generalization of Fröberg's theorem to higher dimensions has been sought in order to classify all square-free monomial ideals with linear resolutions. Such a characterization would also give a description of all square-free monomial ideals which are Cohen-Macaulay.

In this thesis we explore one method of extending Fröberg's result. We generalize the idea of a chordal graph to simplicial complexes and use simplicial homology as a bridge between this combinatorial notion and the algebraic concept of a linear resolution. We are able to give a generalization of one direction of Fröberg's theorem and, in investigating the converse direction, find a necessary and sufficient combinatorial condition for a square-free monomial ideal to have a linear resolution over fields of characteristic 2.

Identiferoai:union.ndltd.org:LACETR/oai:collectionscanada.gc.ca:NSHD.ca#10222/38565
Date07 October 2013
CreatorsConnon, Emma
Source SetsLibrary and Archives Canada ETDs Repository / Centre d'archives des thèses électroniques de Bibliothèque et Archives Canada
LanguageEnglish
Detected LanguageEnglish

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