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  • D. WOODCOCK (a1)


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 [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.



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