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].
Let A be a connected-graded algebra with trivial module k k k , and let B be a graded Ore extension of A. We relate the structure of the Yoneda algebra E A = Ext A k k k k k k to E B . Cassidy and Shelton have shown that when A satisfies their 2 property, B will also be 2 . We prove the converse of this result.
Under certain conditions, a filtration on an augmented algebra A admits a
related filtration on the Yoneda algebra E(A) := Ext_A(K, K). We show that
there exists a bigraded algebra monomorphism from gr E(A) to E_Gr(gr A), where
E_Gr(gr A) is the graded Yoneda algebra of gr A. This monomorphism can be
applied in the case where A is connected graded to determine that A has the K_2
property recently introduced by Cassidy and Shelton.Comment: 14 page
Abstract. We show that the center of a flat graded deformation of a standard Koszul algebra A behaves in many ways like the torus-equivariant cohomology ring of an algebraic variety with finite fixed-point set. In particular, the center of A acts by characters on the deformed standard modules, providing a "localization map." We construct a universal graded deformation of A, and show that the spectrum of its center is supported on a certain arrangement of hyperplanes which is orthogonal to the arrangement coming from the algebra Koszul dual to A. This is an algebraic version of a duality discovered by Goresky and MacPherson between the equivariant cohomology rings of partial flag varieties and Springer fibers; we recover and generalize their result by showing that the center of the universal deformation for the ring governing a block of parabolic category O for gl n is isomorphic to the equivariant cohomology of a Spaltenstein variety. We also identify the center of the deformed version of the "category O" of a hyperplane arrangement (defined by the authors in a previous paper) with the equivariant cohomology of a hypertoric variety.
Abstract. Vatne [12] and Green & Marcos [8] have independently studied the Koszul-like homological properties of graded algebras that have defining relations in degree 2 and exactly one other degree. We contrast these two approaches, answer two questions posed by Green & Marcos, and find conditions that imply the corresponding Yoneda algebras are generated in the lowest possible degrees.
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