Splitting Algebras II: The Cohomology Algebra

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

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