Seminormality and the Cohen -Macaulay property of affine semigroup rings

In the early 1970's, Hochster proved that normal semigroup rings generated by monomials are Cohen-Macaulay. When we weaken normal to seminormal, it is of interest to ask the coincidence of seminormality and the Cohen-Macaulay property of affine semigroup rings.;To solve this problem, three topics are discussed in this thesis. (1) The Cohen-Macaulay property of certain affine semigroup rings; (2) The seminormality of affine semigroup rings; (3) The relation between the Cohen-Macaulay property and seminormality of affine semigroup rings.;The main results of this thesis are: (1) Let S ⊆ Nn be a simplicial affine semigroup. We give a practical criterion to indicate the Cohen-Macaulay property of a simplicial affine semigroup ring in terms of the cardinality of the spanning monomial set of the corresponding semigroup. This is result is extremely useful because it provides a significant algorithm which can be programmed by using Mathematica. (2) We show that, under some hypotheses, the seminormality of an arbitrary affine semigroup ring can be characterized by an extension S' introduced by Goto, Suzuki and Watanabe. (3) We prove that the Cohen-Macaulay property and seminormality coincide under certain hypotheses. If the affine semigroup S contains all interior lattice points of the group in the cone it spans, and the affine semigroup ring R = k[S] is Cohen-Macaulay, then R is seminormal. On the other hand, if the rank of S is less than or equal to 3, S = S', and R is seminormal, then R is Cohen-Macaulay.