Splitting Algebras II: The Cohomology Algebra

Abstract

Gelfand, Retakh, Serconek and Wilson,in [3], defined a graded algebra AΓ 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 AΓ may be strictly smaller that the cohomology algebra (resp. coalgebra) of A Γ .