2016/03/30 by Tyler Kloefkorn, Kloefkorn, Tyler
Mathematics · #Algebraic structures and combinatorial models #Commutative Algebra and Its Applications #Advanced Topics in Algebra
paper · pdf · doi:10.48550/arxiv.1603.09038
To a finite ranked poset \Γ we associate a finite-dimensional graded\nquadratic algebra R_\Γ. Assuming \Γ satisfies a combinatorial\ncondition known as uniform, R\Γ is related to a well-known algebra,\nthe splitting algebra A\Γ. First introduced by Gelfand, Retakh,\nSerconek, and Wilson, splitting algebras originated from the problem of\nfactoring non-commuting polynomials. Given a finite ranked poset \Γ, we\nask: Is R\Γ Koszul? The Koszulity of R\Γ is related to a\ncombinatorial topology property of \Γ called Cohen-Macaulay. Kloefkorn\nand Shelton proved that if \Γ is a finite ranked cyclic poset, then\n\Γ is Cohen-Macaulay if and only if \Γ is uniform and R\Γ\nis Koszul. We define a new generalization of Cohen-Macaulay, weakly\nCohen-Macaulay, and we note that this new class includes posets with\ndisconnected open subintervals. We prove: if \Γ is a finite ranked cyclic\nposet, then \Γ is weakly Cohen-Macaulay if and only if R\Γ is\nKoszul.\n