Green and Marcos [3] call a graded k-algebra δ-Koszul if the corresponding Yoneda algebra Ext(k, k) is finitely generated and Ext(k, k) is zero unless j = δ(i) for some function δ : N→ N. For any integer m ≥ 3 we exhibit a non-commutative quadratic δ-Koszul algebra for which the Yoneda algebra is generated in degrees (1, 1) and (m,m+ 1). These examples answer a question of Green and Marcos. The...

Let A = ⊕ i∈N Ai be a Koszul algebra over a filed K = A0, and *modA the category of finitely generated graded left A-modules. The linearity defect ldA(M) of M ∈ *modA is an invariant defined by Herzog and Iyengar. An exterior algebra E is a Koszul algebra which is the Koszul dual of a polynomial ring. Eisenbud et al. showed that ldE(M) < ∞ for all M ∈ *modE. Improving their result, we show that...

The algebras Qn describe the relationship between the roots and coefficients of a non-commutative polynomial. I.Gelfand, S.Gelfand, and V. Retakh have defined quotients of these algebras corresponding to graphs. In this work we find the Hilbert series of the class of algebras corresponding to the n-vertex path, Pn. We also show this algebra is Koszul. We do this by first looking at class of qua...

One of our main results is a classification all the possible quivers of selfinjective radical cube zero finite dimensional algebras over an algebraically closed field having finite complexity. In the paper [5] we classified all weakly symmetric algebras with support varieties via Hochschild cohomology satisfying Dade’s Lemma. For a finite dimensional algebra to have such a theory of support var...

Background The Koszul cohomology of a line bundle L on an algebraic variety X was introduced by Green in [1]. Koszul cohomology is very closely related to the syzygies of the embedding defined by L (if L is very ample) and is thus related to a host of classical questions. We assume that the reader is already aware of these relations and the main theorems on Koszul cohomology; for a detailed dis...

Recently many researchers have worked on problems connected with various graded algebras associated to an ideal I in a local ring R. Two algebras in particular have received the most attention: the associated graded algebra of Z, gr,( R) = R/Z@ Z/Z’@ ... , and the Rees algebra of Z, defined to be R[Zt]. Brodmann [2] and Goto and Shimoda [9] have studied the local cohomology of R[Zr] for certain...

The aim of this paper is to work out a concrete example as wellas to provide the general pattern of applications of Koszul duality to repre-sentation theory. The paper consists of three parts relatively independent ofeach other.The first part gives a reasonably selfcontained introduction to Koszul ringsand Koszul duality. Koszul rings are certain Z-graded rings with particul...

From symplectic reflection algebras [12], some algebras are naturally introduced. We show that these algebras are non-homogeneous N Koszul algebras. The Koszul property was generalized to homogeneous algebras of degree N > 2 in [6]. In the present paper, the extension of the Koszul property to non-homogeneous algebras is realized through a PBW theorem. This PBW theorem is the generalization to ...

Let A be a noetherian AS regular Koszul quiver algebra (if A is commutative, it is essentially a polynomial ring), and grA the category of finitely generated graded left A-modules. Following Jørgensen, we define the CastelnuovoMumford regularity reg(M•) of a complex M• ∈ D(grA) in terms of the local cohomologies or the minimal projective resolution of M•. Let A be the quadratic dual ring of A. ...