Weakly Cohen-Macaulay posets and a class of finite-dimensional graded quadratic algebras

To a finite ranked poset Γ we associate a finite-dimensional graded quadratic algebra RΓ. Assuming Γ satisfies a combinatorial condition known as uniform, RΓ is related to a well-known algebra, the splitting algebra AΓ. First introduced by Gelfand, Retakh, Serconek, and Wilson, splitting algebras originated from the problem of factoring non-commuting polynomials. Given a finite ranked poset Γ, we ask: Is RΓ Koszul? The Koszulity of RΓ is related to a combinatorial topology property of Γ called Cohen-Macaulay. Kloefkorn and Shelton proved that if Γ is a finite ranked cyclic poset, then Γ is Cohen-Macaulay if and only if Γ is uniform and RΓ is Koszul. We define a new generalization of Cohen-Macaulay, weakly Cohen-Macaulay. This new class includes non-uniform posets and posets with disconnected open subintervals. Using a spectral sequence associated to Γ and the notion of a noncommutative Koszul filtration for RΓ, we prove: if Γ is a finite ranked cyclic poset, then Γ is weakly Cohen-Macaulay if and only if RΓ is Koszul. In addition, we prove that Γ is Cohen-Macaulay if and only if Γ is uniform and weakly Cohen-Macaulay.