We study a finite dimensional quadratic graded algebra R defined from afinite ranked poset. This algebra has been central to the study of thesplitting algebra of the poset, A, as introduced by Gelfand, Retakh, Serconekand Wilson . The algebra A is known to be quadratic when the poset satisfies acombinatorial condition known as uniform, and R is the quadratic dual of anassociated graded algebra of A. We prove that R is Koszul and the poset isuniform if and only if the poset is Cohen-Macaulay. Koszulity of R impliesKoszulity of A. We also show that when R is Koszul, the cohomology of the ordercomplex of the poset can be identified with certain cohomology groups definedinternally to the ring R. Finally, we settle in the negative the long-standingquestion: Does numerically Koszul imply Koszul for algebras of the form R?
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