Noncommutative Koszul algebras from combinatorial topology

Abstract

Abstract. Associated to any uniform finite layered graph Γ there is a noncommutative graded quadratic algebra A(Γ) given by a construction due to Gelfand, Retakh, Serconek and Wilson. It is natural to ask when these algebras are Koszul. Unfortunately, a mistake in the literature states that all such algebras are Koszul. That is not the case and the theorem was recently retracted. We analyze the Koszul property of these algebras for two large classes of graphs associated to finite regular CW complexes, X. Our methods are primarily topological. We solve the Koszul problem by introducing new cohomology groups HX (n, k), generalizing the usual cohomology groups H n (X). Along with several other results, our methods give a new and primarily topological proof of the main result of [12] and [7].

Splitting algebras: Koszul, Cohen–Macaulay and numerically Koszul

“…The first question was answered in the papers [3], [9] and [8]. Some progress was made on the second question in [2], with a subset of those same results appearing in [10]. This paper substantially improves on those results.…”

“…Let Γ be a finite ranked poset with unique minimal element * and strict order <. In the literature this algebra was denoted B(Γ) in [10] and [8] and R(Γ) in [2].…”

“…The paper [2] introduced certain cohomology groups internal to the ring R Γ : H R Γ (n, k) (see 2.6). Our first main theorem shows that these cohomology groups, when R Γ is Koszul, capture all of the cohomology of the topological space associated to the order complex of Γ.…”

Weakly Cohen–Macaulay posets and a class of finite-dimensional graded quadratic algebras

“…We now turn our attention to R Γ form Definition 1.1. Cassidy, Phan, and Shelton proved the following lemma in [2]. This lemma is a very powerful tool; we will use it repeatedly and without further comment.…”

“…We remind the reader of the following fact from Section 3 in [2]: for a uniform Γ, the Koszulity of R Γ is equivalent to a condition on right annihilators of certain elements of R Γ . In this section, we establish a more general result.…”