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

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 . . . . . . . . . . . . .