Convex geometric, homological and combinatorial properties of graded ideals [Elektronische Ressource] / von Pooja Singla

Description

Convex–geometric, homologicaland combinatorial properties ofgraded idealsVom Fachbereich Mathematik derUniversit¨at Duisburg–Essenzur Erlangung des akademischen Grades einesDr. rer. nat.genehmigte DissertationvonPooja SinglaausIndienReferent: Prof. Dr. HerzogKoreferent: Prof. Dr. WelkerDezember 2007Dedicated to my parentsAcknowledgmentsI would like to express my deepest gratitude to my advisor Prof. Ju¨rgen Herzog forhis continuous guidance, tremendous support and encouragement throughout thepreparation of my thesis.My heartiest thanks to Prof. Jugal Verma, the advisor of my master thesis, whointroduced me to Commutative Algebra and inspired me in all ways.I also wish to thank Prof. Hibi, Prof. Trung and Satoshi Murai with whom Idiscussed mathematics during their visits to the department of mathematics overthe last years.Many thanks to all my colleagues, my office mates and my friends in Essen.Lastly, I thank my family for their encouragement and moral support.ContentsIntroduction 11 Minimal monomial reductions and the reduced fiber ring of anextremal ideal 71.1 Some preliminaries on the convex geometry of monomial ideals . . . . 81.2 Minimal monomial reduction ideals . . . . . . . . . . . . . . . . . . . 10m1.3 A description of the faces of conv(I ) . . . . . . . . . . . . . . . . . . 121.4 The structure of the reduced fiber ring of an extremal ideal . . . . . . 141.5 On the reduction number of a monomial ideal . . . . . . . . . . . . .

Sujets

• Mathematics

Informations

• Publié par: universitat_duisburg-essen
• Publié le: 01 janvier 2007
• Nombre de lectures: 17
• Langue: English

Extrait

Convex–geometric, homological and combinatorial properties of graded ideals

Vom Fachbereich Mathematik der Universita¨tDuisburg–Essen zur Erlangung des akademischen Grades eines Dr. rer. nat.

genehmigte Dissertation

von

Pooja Singla

aus Indien

Referent: Prof. Dr. Herzog Koreferent: Prof. Dr. Welker

Dezember 2007

Dedicated to my parents

Acknowledgments

IwouldliketoexpressmydeepestgratitudetomyadvisorProf.Ju¨rgenHerzogfor his continuous guidance, tremendous support and encouragement throughout the preparation of my thesis. My heartiest thanks to Prof. Jugal Verma, the advisor of my master thesis, who introduced me to Commutative Algebra and inspired me in all ways. I also wish to thank Prof. Hibi, Prof. Trung and Satoshi Murai with whom I discussed mathematics during their visits to the department of mathematics over the last years. Many thanks to all my colleagues, my office mates and my friends in Essen. Lastly, I thank my family for their encouragement and moral support.

Introduction

Contents

1 Minimal monomial reductions and the reduced fiber ring of an extremal ideal 1.1 Some preliminaries on the convex geometry of monomial ideals . . . . 1.2 Minimal monomial reduction ideals . . . . . . . . . . . . . . . . . . . 1.3 A description of the faces of conv(I m . . . . . . . . . . . . . . . . . ) . 1.4 The structure of the reduced fiber ring of an extremal ideal . . . . . . 1.5 On the reduction number of a monomial ideal . . . . . . . . . . . . .

2

3

4

Graded Betti numbers and the regularity function 2.1 Diagonal submodules . . . . . . . . . . . . . . . . . . 2.2 Graded Betti numbers of powers of ideals . . . . . . . 2.3 The regularity of power products of graded ideals . .

. . .

Rigidity of linear strands and generic initial ideals 3.1 An upper bound for the graded Betti numbers . . . . . 3.2 Graded rigidity of resolutions and linear components . 3.3 The Cartan–complex and generic annihilator numbers . 3.4 Rigidity of resolutions over the exterior algebra . . . . 3.5 Linear components and graded Betti numbers . . . . . 3.6 The cancellation principle . . . . . . . . . . . . . . . .

Linear balls and the multiplicity conjecture 4.1 The multiplicity conjecture . . . . . . . . . . . 4.2 Determinantal ideals . . . . . . . . . . . . . . 4.3 Polarization of the powers of a maximal ideal

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Bibliography

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