Let R be a noetherian ring andM a finite R-module. With a linear form χ onM one associates the Koszul complex K(χ). IfM is a free module, then the homology of K(χ) is well-understood, and in particular it is grade sensitive with respect to Imχ. In this note we investigate the case of a module M of projective dimension 1 (more precisely, M has a free resolution of length 1) for which the first n...

Abstract We study a semisimple extension of Takiff superalgebra, which turns out to have remarkably rich representation theory. determine the blocks in both finite-dimensional and BGG module categories also classify Borel subalgebras. further compute all groups between two simple objects prove that non-principal category are Koszul.

We introduce here the notion of Koszul duality for monoids in the monoidal category of species with respect to the ordinary product. To each Koszul monoid we associate a class of Koszul algebras in the sense of Priddy, by taking the corresponding analytic functor. The operad AM of rooted trees enriched with a monoid M was introduced by the author many years ago. One special case of that is the ...

Koszul pairs were introduced in [arXiv:1011.4243] as an instrument for the study of Koszul rings. In this paper, we continue the enquiry of such pairs, focusing on the description of the second component, as a follow-up of the study in [arXiv:1605.05458]. As such, we introduce Koszul corings and prove several equivalent characterizations for them. As applications, in the case of locally finite ...

The N -Koszul algebras are N -homogeneous algebras satisfying a homological property. These algebras are characterised by their Koszul complex: an N -homogeneous algebra is N -Koszul if and only if its Koszul complex is acyclic. Methods based on computational approaches were used to prove N -Koszulness: an algebra admitting a side-confluent presentation is N -Koszul if and only if the extra-con...

The concept of Koszul differential graded algebra (Koszul DG algebra) is introduced. Koszul DG algebras exist extensively, and have nice properties similar to the classic Koszul algebras. A DG version of the Koszul duality is proved. When the Koszul DG algebra A is AS-regular, the Ext-algebra E of A is Frobenius. In this case, similar to the classical BGG correspondence, there is an equivalence...

We investigate relations between the properties of an algebra and its varieties of finite-dimensional module structures, on the example of the Jordan plane R = k〈x, y〉/(xy − yx− y). Complete description of irreducible components of the representation variety mod(R,n) obtained for any dimension n, it is shown that the variety is equidimensional. The influence of the property of the non-commutati...

Let M be a finite module over a noetherian ring R with a free resolution of length 1. We consider the generalized Koszul complexes Cλ̄(t) associated with a map λ̄ : M → H into a finite free R-module H (see [IV], section 3), and investigate the homology of Cλ̄(t) in the special setup when grade IM = rankM = dimR. (IM is the first non-vanishing Fitting ideal of M .) In this case the (interesting) ho...

A contravariant functor is constructed from the stable projective homotopy theory of finitely generated graded modules over a finite-dimensional algebra to the derived category of its Yoneda algebra modulo finite complexes of modules of finite length. If the algebra is Koszul with a noetherian Yoneda algebra, then the constructed functor is a duality between triangulated categories. If the alge...