Hilbert functions of Cohen— Macaulay integral domains and chain conditions of finite partially ordered sets

AbstractLet A = A0 ⊕ A1 ⊕ ⋯ be a commutative graded ring such that (i) A0 = k a field. (ii) A = k[A1] and (iii) dimkA1 < ∞. It is a fundamental fact that the formal power series ∑∗n=0(dimλ An)λn is of the form (h0 + h1λ + ⋯ + h8λx)/(1-λ) dim A, where each h, is an integer. We are interested in the sequence (h0,h1,…,hx) called tL. h-vector of A, when A is a Cohen-Macaulay integral domain. In this paper, first, in Section 1, we summarize the basic information on the h-vector of A which can be obtained by investigating the behavior of the canonical module KA of A. Secondly, in Section 2, we apply the abstract theory in Section 1 to the combinatorial problem of finding a characterization of the so-called w-vectors of finite partially ordered sets and obtain linear and nonlinear inequalities for the w-vector of a finite partially ordered set which satisfies a certain chain condition