2012/08/10 by Brad Shelton, Shelton, Brad
Mathematics · #05E15 #16W50 #Advanced Combinatorial Mathematics #Algebraic structures and combinatorial models #Commutative Algebra and Its Applications #FOS: Mathematics #Rings and Algebras (math.RA) #math.RA #msc:05E15 #msc:16W50
paper · pdf · doi:10.48550/arxiv.1208.2202
16 pages, 1 figure
arxiv created 2012/08/10 · openalex publication_date 2012/08/10 · arxiv updated 2012/08/13 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
Gelfand, Retakh, Serconek and Wilson, in \citeGRSW, 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.