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

To a finite ranked poset Gamma we associate a finite-dimensional graded quadratic algebra R-Gamma. Assuming Gamma satisfies a combinatorial condition known as uniform, Rr is related to a well-known algebra, the splitting algebra A(Gamma). First introduced by Gelfand, Retakh, Serconek, and Wilson, splitting algebras originated from the problem of factoring non-commuting polynomials. Given a finite ranked poset Gamma, we ask: Is R-Gamma Koszul? The Koszulity of R-Gamma is related to a combinatorial topology property of Gamma called Cohen-Macaulay. Kloefkorn and Shelton proved that if Gamma is a finite ranked cyclic poset, then Gamma is Cohen-Macaulay if and only if Gamma is uniform and R-Gamma 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 Gamma and the notion of a noncommutative Koszul filtration for R-Gamma, we prove: if Gamma is a finite ranked cyclic poset, then Gamma is weakly Cohen-Macaulay if and only if R-Gamma is Koszul. In addition, we prove that Gamma is Cohen-Macaulay if and only if Gamma is uniform and weakly Cohen Macaulay. (C) 2017 Elsevier Inc. All rights reserved.