  We study a finite dimensional quadratic graded algebra R defined from a
finite ranked poset. This algebra has been central to the study of the
splitting algebra of the poset, A, as introduced by Gelfand, Retakh, Serconek
and Wilson . The algebra A is known to be quadratic when the poset satisfies a
combinatorial condition known as uniform, and R is the quadratic dual of an
associated graded algebra of A. We prove that R is Koszul and the poset is
uniform if and only if the poset is Cohen-Macaulay. Koszulity of R implies
Koszulity of A. We also show that when R is Koszul, the cohomology of the order
complex of the poset can be identified with certain cohomology groups defined
internally to the ring R. Finally, we settle in the negative the long-standing
question: Does numerically Koszul imply Koszul for algebras of the form R?
