Contributions to the theory of normal affine semigroup rings and Ulrich modules of rank one over determinantal rings [Elektronische Ressource] / von Attila Wiebe

Contributions to the theory ofnormal affine semigroup ringsandUlrich modules of rank oneover determinantal ringsDissertationdem Fachbereich Mathematikder Universit¨ at Duisburg-Essenzur Erlangung desDoktorgrades (Dr. rer. nat)vorgelegt im April 2006vonAttila Wiebeaus BochumTag der mundlic¨ hen Prufung:¨ 30. Juni 2006Vorsitzender der Prufungsk¨ ommission: Prof. Dr. Rudiger¨ G¨ obelGutachter: Prof. Dr. Winfried BrunsProf. Dr. Jurgen¨ HerzogFor my wife1ContentsIntroduction 21. The Rees algebra of a normal affine semigroup ring 41.1. Affine semigroup rings 41.2. A short excursion into convex geometry 61.3. The bottom of an affine semigroup 91.4. The integral closure of an ideal 101.5. The associated graded ring of an affine semigroup ring 121.6. Normality of the Rees algebra 141.7. Cohen-Macaulayness of the Rees algebra 211.8. The special case of hypersurface rings 262. On the type of a simplicial normal affine semigroup ring 302.1. Preparations 302.2. The case of dimension 2 322.3. The case of 3 323. Ulrich modules of rank one over determinantal rings 383.1. Ulrich modules 383.2. Determinantal rings 413.3. The existence of Ulrich modules of rank one 43References 522IntroductionThe theory of affine semigroup rings and the theory of determinantal rings areappealing and vital branches of present-day commutative algebra. In the investi-gation of these rings, both geometric and combinatoric aspects play an importantrole.