On Algebras Associated to Finite Ranked Posets and Combinatorial Topology: The Koszul, Numerically Koszul and Cohen-Macaulay Properties
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This dissertation studies new connections between combinatorial topology and homological algebra. To a finite ranked poset Γ we associate a finite-dimensional quadratic graded 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 a standard question in homological algebra: Is RΓ Koszul? The Koszulity of RΓ is related to a combinatorial topology property of Γ known as Cohen-Macaulay. One of the main theorems of this dissertation is: 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 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 Γ is a finite ranked cyclic poset, then Γ is weakly Cohen-Macaulay if and only if RΓ is Koszul. Finally, we address the notion of numerical Koszulity. We show that there exist algebras RΓ that are numerically Koszul but not Koszul and give a general construction for such examples. This dissertation includes unpublished co-authored material.