Koszulity and Point Modules of Finitely Semi-Graded Rings and Algebras

In this paper, we investigate the Koszul behavior of finitely semi-graded algebras by the distributivity of some associated lattice of ideals. The Hilbert series, the Poincaré series, and the Yoneda algebra are defined for this class of algebras. Moreover, the point modules and the point functor are introduced for finitely semi-graded rings. Finitely semi-graded algebras and rings include many important examples of non- N -graded algebras coming from mathematical physics that play a very important role in mirror symmetry problems, and for these concrete examples, the Koszulity will be established, as well as the explicit computation of its Hilbert and Poincaré series.

Modules with pure resolutions over Koszul algebras revisited

I construct a Koszul algebra A and a finitely generated graded A-module M that together form a counterexample to a recently published claim. M is generated in degree 0 and has a pure resolution, and the graded Jacobson radical of the Yoneda algebra of A does not annihilate the Ext module of M, but nonetheless M is not a Koszul module.

THE HOMOGENISED ENVELOPING ALGEBRA OF THE LIE ALGEBRA sℓ(2,ℂ)

In this paper, we study the homogenised algebra B of the enveloping algebra U of the Lie algebra sℓ(2,ℂ). We look first to connections between the category of graded left B-modules and the category of U-modules, then we prove B is Koszul and Artin–Schelter regular of global dimension four, hence its Yoneda algebra B! is self-injective of radical five zeros, and the structure of B! is given. We describe next the category of homogenised Verma modules, which correspond to the lifting to B of the usual Verma modules over U, and prove that such modules are Koszul of projective dimension two. It was proved in Martínez-Villa and Zacharia (Approximations with modules having linear resolutions, J. Algebra266(2) (2003), 671–697)] that all graded stable components of a self-injective Koszul algebra are of type ZA∞. Here, we characterise the graded B!-modules corresponding to the Koszul duality to homogenised Verma modules, and prove that these are located at the mouth of a regular component. In this way we obtain a family of components over a wild algebra indexed by ℂ.

MODULES WITH PURE RESOLUTIONS OVER KOSZUL ALGEBRAS

Let A be a Koszul algebra and M a finitely generated graded A-module. Suppose that M is generated in degree 0 and has a pure resolution. We prove that, if rℰ(M) ≠ 0 then M is Koszul; and if in addition M is not projective, then the converse is true as well, where r denotes the graded Jacobson radical of the Yoneda algebra [Formula: see text] of A, and [Formula: see text] denotes the Ext module of M.

ON THE EXISTENCE OF A DERIVED EQUIVALENCE BETWEEN A KOSZUL ALGEBRA AND ITS YONEDA ALGEBRA

In this paper, we study the derived categories of a Koszul algebra and its Yoneda algebra to determine when those categories are triangularly equivalent. We prove that the simply connected Koszul algebras are derived equivalent to their Yoneda algebras. We have considered discrete Koszul algebras and we gave necessary and sufficient conditions for those Koszul algebras to be derived equivalent to their Yoneda algebras. We also study the class of Koszul algebras which are derived equivalent to hereditary algebras. For the case where the hereditary algebra is tame, we characterized the derived equivalence between those Koszul algebras and their Yoneda algebras.

HIGHER KOSZUL DUALITY FOR ASSOCIATIVE ALGEBRAS

We present a unifying framework for the key concepts and results of higher Koszul duality theory for N-homogeneous algebras: the Koszul complex, the candidate for the space of syzygies and the higher operations on the Yoneda algebra. We give a universal description of the Koszul dual algebra under a new algebraic structure. For that we introduce a general notion: Gröbner bases for algebras over non-symmetric operads.