## Classifications of Cohen-Macaulay modules - The base ring associated to a transversal polymatroid

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Ştefan, Alin (2008)
• Subject: Mathematics - Combinatorics | 13P10 (Primary) 13H10, 13D40, 13A02 (Secondary) | Mathematics - Commutative Algebra

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 \cite{HH} in 2002.
• References (12)
12 references, page 1 of 2

1 Background. 6 1.1 A short excursion into convex geometry. . . . . . . . . . . . . . . . . . . . 6 1.2 Affine semigroup rings. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9 1.3 Discrete polymatroids. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11 1.4 The Ehrhart ring and the base ring of a discrete polymatroid. . . . . . . . 18

2 The type of the base ring associated to a transversal polymatroid 21 2.1 Cones of dimension n with n + 1 facets. . . . . . . . . . . . . . . . . . . . 21 2.2 The type of base ring associated to transversal polymatroids with the cone of dimension n with n + 1 facets. . . . . . . . . . . . . . . . . . . . . . . . 26 2.3 Ehrhart function. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 4 A remark on the Hilbert series of transversal polymatroids 53 4.1 Segre product and the base ring associated to a transversal polymatroid. . 53 4.2 Hilbert series. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 56 Bibliography. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 63

[3] W. Bruns, J. Herzog, Cohen-Macaulay rings, Revised Edition, Cambridge, 1997.

[4] W. Bruns, J. Gubeladze, Polytopes, rings and K-theory, preprint.

[25] A. S¸tefan, The Facets Cone Associated To Some Classes Of Transversal Polymatroids, Analele S¸tiin¸tifice ale Universita¸tii ”Ovidius” Constan¸ta Seria Matematica volumul XV (2007), fascicola 1, 139-158.

[27] E. Miller, B. Sturmfels, Combinatorial commutative algebra, Graduate Texts in Mathematics 227, Springer-Verlag, New-York, 2005.

[28] R. Villarreal, Monomial Algebras, Marcel Dekker, New-York, 2001.

[29] M. Vl˘adoiu, Discrete polymatroids, An. S¸t. Univ. Ovidius, Constan¸ta, 14 (2006), 89- 112.

[30] M. Vl˘adoiu, Equidimensional and unmixed ideals of Veronese type, to appear in Communications in Alg., arXiv:math. AC/0611326.

[31] G. M. Ziegler, Lecture on Polytopes, Graduate Texts in Mathematics 152, SpringerVerlag, New-York, 1995.

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