Splitting algebras: Koszul, Cohen-Macaulay and numerically Koszul

We study a finite dimensional quadratic graded algebra R-Gamma defined from a finite ranked poset P. This algebra has been central to the study of the splitting algebras A(Gamma) introduced by Gelfand, Retakh, Serconek and Wilson [4]. Those algebras are known to be quadratic when Gamma satisfies a combinatorial condition known as uniform. A central question in this theory has been: when are the algebras Koszul? We prove that R-Gamma is Koszul and P is cyclic and uniform if and only if the poset is Cohen-Macaulay. We also show that the cohomology of the order complex of Gamma can be identified with certain cohomology groups defined internally to the ring R-Gamma, H-R Gamma (n, 0) (introduced in [2]) whenever Gamma is Cohen-Macaulay. Finally, we settle in the negative the long-standing question: Does numerically Koszul imply Koszul for algebras of the form R-Gamma. (C) 2014 Elsevier Inc. All rights reserved.