Lefschetz elements for Stanley-Reisner rings and annihilator numbers [Elektronische Ressource] / vorgelegt von Martina Kubitzke

Martina KubitzkeLefschetz Elements forStanley-Reisner RingsandAnnihilator NumbersLefschetz Elements forStanley-Reisner RingsandAnnihilator NumbersDissertationzurErlangung des Doktorgradesder Naturwissenschaften(Dr. rer. nat.)demFachbereich Mathematik und Informatikder Philipps-Universitat Marburgvorgelegt vonMartina Kubitzkegeboren in Frankfurt am MainMarburg 2009AbstractThis thesis is composed of three big parts. In the first two chapters we provide the basicdefinitions and facts which are needed in the subsequent chapters. Chapter 1 is concernedwith basic algebraic principles whereas in the second chapter we treat simplicial complexes.The actual results of this thesis are presented in Chapter 3 through 5. In particular, wededicate Chapters 3 and 4 to the g-conjecture, the g-theorem, further related results as wellas the Lefschetz property for barycentric subdivisions of shellable simplicial complexes.In Chapter 5, which constitutes the second main topic, we consider the symmetric andthe exterior depth of finitely generated modules as well as the symmetric and the exteriorannihilator numbers. In the following we dwell on the two main topics in more detail.One of the most classical and most studied problems in combinatorial commutative alge-bra and discrete geometry is the characterization of f -vectors of special classes of simplicialcomplexes.