Related papers: Koszul pairs and applications

Given an acyclic twisting cochain $\pi:C\to A$, from a curved dg coalgebra $C$ to a dg algebra $A$, we show that the associated twisted hom complex $\mathrm{Hom}^\pi_k(C,A)$ has cohomology equal to the Hochschild cohomology of $A$, as a…

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…

The Koszul homology algebra of a commutative local (or graded) ring $R$ tends to reflect important information about the ring $R$ and its properties. In fact, certain classes of rings are characterized by the algebra structure on their…

We define generalized Koszul modules and rings and develop a generalized Koszul theory for $\mathbb{N}$-graded rings with the degree zero part noetherian semiperfect. This theory specializes to the classical Koszul theory for graded rings…

In our preceding paper we have introduced the notion of an $s$-homogeneous triple. In this paper we use this technique to study connected $s$-homogeneous algebras with two relations. For such algebras, we describe all possible pairs…

We study the so-called weakly Koszul modules and characterise their Koszul duals. We show that the (adjusted) associated graded module of a weakly Koszul module exactly determines the homology modules of the Koszul dual. We give an example…

We give a complete picture of the interaction between Koszul and Ringel dualities for quasi-hereditary algebras admitting linear tilting (co)resolutions of standard and costandard modules. We show that such algebras are Koszul, that the…

Let A and A! be dual Koszul algebras. By Positselski a filtered algebra U with gr U = A is Koszul dual to differential graded algebra (A!,d). We relate the module categories of this dual pair by a tensor-Hom adjunction. This descends to…

Let $k$ be a field. Let $A$ and $B$ be connected $N$-graded $k$-algebras. Let $C$ denote a twisted tensor product of $A$ and $B$ in the category of connected $N$-graded $k$-algebras. The purpose of this paper is to understand when $C$…

In this article we discuss two different but related results on Hochschild (co)homology and the theory of Koszul duality. On the one hand, we prove essentially that the Tamarkin-Tsygan calculus of an Adams connected augmented dg algebra and…

Let C be a semidualizing complex over a noetherian local ring A. If there exists a local homomorphism with source A satisfying some homological properties, then C is dualizing.

Let $a$ and $b$ be two integers such that $2\le a<b$. In this article we define the notion of $(a,b)$-Koszul algebra as a generalization of $N$-Koszul algebras. We also exhibit examples and we provide a minimal graded projective resolution…

Let $k$ be a field and $R$ a standard graded $k$-algebra. We denote by $\operatorname{H}^R$ the homology algebra of the Koszul complex on a minimal set of generators of the irrelevant ideal of $R$. We discuss the relationship between the…

In this paper we prove that if G is a connected, simply-connected, semi-simple algebraic group over an algebraically closed field of sufficiently large characteristic, then all the blocks of the restricted enveloping algebra (Ug)_0 of the…

We prove in a unifying way several equivalent descriptions of Koszul rings, some of which being well known in the literature. Most of them are stated in terms of coring theoretical properties of $\Tor_n^A(R,R)$. As an application of these…

Let $H$ be a numerical semigroup. We give effective bounds for the multiplicity $e(H)$ when the associated graded ring $\operatorname{gr}_\mathfrak{m} K[H]$ is defined by quadrics. We classify Koszul complete intersection semigroups in…

Let $\check{C}_{\underline{x}}$ denote the \v{C}ech complex with respect to a system of elements $\underline{x} = x_1,\ldots,x_r$ of a commutative ring $R$. We construct a bounded complex $\mathcal{L}_{\underline{x}}$ of free $R$-modules…

The aim of this article is to study the Ext ring associated to a Koszul $R$-ring and to use it to provide further characterisations of the former. As such, for $R$ being a semisimple ring and $A$ a graded Koszul $R$-ring, we will prove that…

We investigate compatibility of gradings for an almost Koszul or Koszul algebra $R$ that is also the higher preprojective algebra $\Pi_{n+1}(A)$ of an $n$-hereditary algebra $A$. For an $n$-representation finite algebra $A$, we show that…

We prove that on a certain class of smooth complex varieties (those with "affine even stratifications"), the category of mixed Hodge modules is "almost" Koszul: it becomes Koszul after a few unwanted extensions are eliminated. We also give…