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Journal of the London Mathematical Society 1998 57(2):398-410; doi:10.1112/S0024610798005717
© 1998 by London Mathematical Society
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© The London Mathematical Society

Cohen–Macaulay Complexes and Koszul Rings

D. Woodcock

Mathematics Department, City University Northampton Square, London EC1V 0HB

Received 15 March 1994.

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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