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, Björner, Garsia and Stanley  or Budach, Graw, Meinel and Waack . It is known (see, for example, Cibils , Gerstenhaber and Schack ) 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.