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 $R_Γ$. Assuming $Γ$ satisfies a combinatorial condition known as uniform, $R_Γ$ is related to a well-known algebra, the splitting algebra $A_Γ$. 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 $R_Γ$ Koszul? The Koszulity of $R_Γ$ 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 $R_Γ$ is Koszul. We define a new generalization of Cohen-Macaulay, weakly Cohen-Macaulay, and we note that this new class includes posets with disconnected open subintervals. We prove: if $Γ$ is a finite ranked cyclic poset, then $Γ$ is weakly Cohen-Macaulay if and only if $R_Γ$ is Koszul.

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