On algebras associated to finite ranked posets and combinatorial topology: The Koszul, numerically Koszul and Cohen-Macaulay properties

This dissertation studies new connections between combinatorial topology and homological algebra. To a finite ranked poset Gamma we associate a finite-dimensional quadratic graded algebra RGamma. Assuming Gamma satisfies a combinatorial condition known as uniform, RGamma is related to a well-known algebra, the splitting algebra AGamma. 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 a standard question in homological algebra: Is RGamma Koszul? The Koszulity of RGamma is related to a combinatorial topology property of Gamma known as Cohen-Macaulay. One of the main theorems of this dissertation is: If Gamma is a finite ranked cyclic poset, then Gamma is Cohen-Macaulay if and only if Gamma is uniform and RGamma is Koszul.;We also define a new generalization of Cohen-Macaulay: weakly Cohen-Macaulay. The class of weakly Cohen-Macaulay finite ranked posets includes posets with disconnected open subintervals. We prove: if Gamma is a finite ranked cyclic poset, then Gamma is weakly Cohen-Macaulay if and only if RGamma is Koszul. Finally, we address the notion of numerical Koszulity. We show that there exist algebras RGamma that are numerically Koszul but not Koszul and give a general construction for such examples. This dissertation includes unpublished co-authored material.