Article ID | Journal | Published Year | Pages | File Type |
---|---|---|---|---|
5771962 | Journal of Algebra | 2017 | 23 Pages |
Abstract
To a finite ranked poset Î we associate a finite-dimensional graded quadratic algebra RÎ. Assuming Î satisfies a combinatorial condition known as uniform, RÎ is related to a well-known algebra, the splitting algebra AÎ. First introduced by Gelfand, Retakh, Serconek, and Wilson, splitting algebras originated from the problem of factoring non-commuting polynomials. Given a finite ranked poset Î, we ask: Is RÎ Koszul? The Koszulity of RÎ is related to a combinatorial topology property of Î called Cohen-Macaulay. Kloefkorn and Shelton proved that if Î is a finite ranked cyclic poset, then Î is Cohen-Macaulay if and only if Î is uniform and RÎ 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 Î and the notion of a noncommutative Koszul filtration for RÎ, we prove: if Î is a finite ranked cyclic poset, then Î is weakly Cohen-Macaulay if and only if RÎ is Koszul. In addition, we prove that Î is Cohen-Macaulay if and only if Î is uniform and weakly Cohen-Macaulay.
Related Topics
Physical Sciences and Engineering
Mathematics
Algebra and Number Theory
Authors
Tyler Kloefkorn,