Let S be an affine semigroup, i.e., a finitely generated submonoid of the additive monoid N , where N is the set of nonnegative integers and r is w x some number, and let k S denote the semigroup ring over S, where k is a w x field. Then k S is a homomorphic image of some polynomial ring over k. Let I be an ideal of A such that the ring ArI is canonically isomorphic to w x w x k S . It is natural to examine I to investigate the properties of k S , for example, Cohen]Macaulay and Buchsbaum properties. Kamoi proved that, under the conditions that S is simplicial and that the height of I is 2, w x Ž . k S is Cohen]Macaulay if and only if m I F 3, where m denotes the w x least number of generators 10 . Independently, there are many results w x Ž . concerning the relationship between the properties of k S and m I , 3 Ž w x. when it is the coordinate ring of the monomial curve in P e.g., 1]4, 9 . In this paper, we will generalize these results uniformly. We only set the assumption that the height of I is 2. In Section 1, we will prepare the notation of monoid rings. Monoid rings are defined for submodules of Z , where Z is the ring of integers, and w x studied in 7 . An affine semigroup ring is a monoid ring. In Section 2, we will find a minimal generating system of the defining ideal of the monoid ring associated with the submodule of Z r of rank 2, when it is positively graded and we will give a minimal free resolution of it. Consequently, we can prove that
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Open Access
Jan 1, 1997
Klaus G Fischer + 2
