Splitting algebras: Koszul, Cohen–Macaulay and numerically Koszul

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