A ug 2 00 8 Classifications of Cohen - Macaulay modules - The base ring associated

In this thesis, we focus on the study of the base rings associated to some transversal polymatroids. A transversal polymatroid is a special kind of discrete polymatroid. Discrete polymatroids were introduced by Herzog and Hibi [16] in 2002. The thesis is structured in four chapters. Chapter 1 starts with a short excursion into convex geometry. Next, we look at some properties of affine semigroup rings. We recall some basic definitions and known facts about semigroup rings. Next we give a brief introduction to matroids and discrete polymatroids. Finally, we remind some properties of the base rings associated to discrete polymatroids. These properties will be needed in the next chapters of the thesis. Chapter two is devoted to the study of the canonical module of the base ring associated to a transversal polymatroid. We determine the facets of the polyhedral cone generated by the exponent set of monomials defining the base ring. This allows us to describe the canonical module in terms of the relative interior of the cone. Also, this would allow one to compute the a− invariant of the base ring. Since the base ring associated to a discrete polymatroid is normal it follows that Ehrhart function is equal with Hilbert function and knowing the a− invariant we can very easy get its Hilbert series. We end this chapter with the following open problem Open Problem: Let n ≥ 4, Ai ⊂ [n] for any 1 ≤ i ≤ n and K[A] be the base ring associated to the transversal polymatroid presented by A = {A1, . . . , An}. If the Hilbert series is: HK[A](t) = 1 + h1 t+ . . .+ hn−r t n−r (1− t)n , then we have the following: 1) If r = 1, then type(K[A]) = 1 + hn−2 − h1. 2) If 2 ≤ r ≤ n, then type(K[A]) = hn−r. In chapter three we study intersections of Gorenstein base rings. These are also Gorenstein rings and we are interested when the intersections of Gorenstein base rings are the base rings associated to some transversal polymatroids. More precisely, we give necessary and sufficient conditions for the intersection of two base rings to be still a base ring of a transversal polymatroid. In chapter four we study when the transversal polymatroids presented by A = {A1, A2, . . . , Am} with |Ai| = 2 have the base ring K[A] Gorenstein. Using Worpitzky 3 identity, we prove that the numerator of the Hilbert series has the coefficients Eulerian numbers and from [1] the Hilbert series is unimodal. We acknowledge the support provided by the Computer Algebra Systems NORMALIZ [5] and SINGULAR [8] for the extensive experiments which helped us to obtain some of the results in this thesis.