Splitting Algebras Ii: the Cohomology Algebra
نویسنده
چکیده
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 AΓ. We calculate the cohomology algebra (and coalgebra) of AΓ explicitly. As a corollary to this calculation we have a proof that AΓ 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 AΓ.
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