Cohen-Macaulay Complexes and Koszul Rings

Abstract

Throughout this paper k denotes a fixed commutative ground ring. A Cohen–Macaulay complex is a finite simplicial complex satisfying a certain homological vanishing condition. These complexes have been the subject of much research; introductions can be found in, for example, Bjorner, Garsia and Stanley [6] or Budach, Graw, Meinel and Waack [7]. It is known (see, for example, Cibils [8], Gerstenhaber and Schack [10]) that there is a strong connection between the (co)homology of an arbitrary simplicial complex and that of its incidence algebra. We show how the Cohen–Macaulay property fits into this picture, establishing the following characterization.A pure finite simplicial complex is Cohen–Macaulay over k if and only if the incidence algebra over k of its augmented face poset, graded in the obvious way by chain lengths, is a Koszul ring.