Cohen-Macaulay ordered sets

The purpose of this paper is to introduce a new kind of partially ordered set: the Cohen-Macaulay poset. It is now known that this concept provides some interesting connections among Algebraic Topology, Combinatorics, Commutative Algebra and Homological Algebra, and numerous individuals have contributed to the theory. Some of these are Stanley [35, 36, 371, Reisner [26], Hochster [21] and Garsia [17]. The notion of a Cohen-Macaulay poset originated in the author’s thesis [2], and many of the results of this paper are also there in some form. The original motivation for introducing this concept was to provide a reasonable setting for the results of [I] and to find techniques for proving unimodality theorems. The Rank Selection Theorem (5.4) h a d much to do with this. At the time we referred to these posets as Folkman posets because of Folkman’s work in [16]. The term “Cohen-Macaulay” was later suggested by Kempf, who pointed out the relationship with the theory of Local Cohomology as, for example, in [22]. The basic tool for proving our results is the theory of homology of diagrams on posets. Diagrams, even without homology, are related to certain purely combinatorial constructions. For an example of this see [4]. By using diagrams in more sophisticated ways one can prove some quite interesting combinatorial theorems, as was done for example in [6] using results from [3]. Although we have consistently used poset homology to prove the results in this paper, one could also prove them using ring theory methods. In a joint paper with Garsia [7], the latter approach is employed. The fact that one can define Cohen-Macaulay posets using either homology theory or ring theory is a consequence of a remarkable theorem of Reisner [26]. An “elementary” proof of this important theorem appears in [7]. One of the most dramatic applications of Cohen-Macaulay posets (or more precisely of Cohen-Macaulay complexes) is the proof by Stanley [35] of the