Cohen-Macaulay Complexes and Koszul Rings

Throughout this paper k denotes a fixed commutative ground ring.A Cohen–Macaulay complex is a finite simplicial complexsatisfying a certain homological vanishing condition. Thesecomplexes have been the subject of much research; introductionscan be found in, for example, Björner, Garsia and Stanley6] or Budach, Graw, Meinel and Waack 7]. It is known (see,for example, Cibils 8], Gerstenhaber and Schack 10]) thatthere is a strong connection between the (co)homology of anarbitrary simplicial complex and that of its incidence algebra.We show how the Cohen–Macaulay property fits into thispicture, establishing the following characterization. A pure finite simplicial complex is Cohen–Macaulay overk if and only if the incidence algebra over k of its augmentedface poset, graded in the obvious way by chain lengths, is aKoszul ring.