Splitting Algebras II: The Cohomology Algebra

Gelfand, Retakh, Serconek and Wilson, in \cite{GRSW}, defined a graded algebra $A_\Gamma$ attached to any finite ranked poset $\Gamma$ - a generalization of the universal algebra of pseudo-roots of noncommutative polynomials. This algebra has since come to be known as the splitting algebra of $\Gamma$. The splitting algebra has a secondary filtration related to the rank function on the poset and the associated graded algebra is denoted here by $A'_\Gamma$. We calculate the cohomology algebra (and coalgebra) of $A'_\Gamma$ explicitly. As a corollary to this calculation we have a proof that $A'_\Gamma$ is Koszul (respectively quadratic) if and only if $\Gamma$ is Cohen-Macaulay (respectively uniform). We show by example that the cohomology algebra (resp. coalgebra) of $A_\Gamma$ may be strictly smaller that the cohomology algebra (resp. coalgebra) of $A'_\Gamma$.