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A Priddy-type Koszulness criterion for non-locally finite algebras

Maurizio Brunetti, Adriana Ciampella (2007)

Colloquium Mathematicae

A celebrated result by S. Priddy states the Koszulness of any locally finite homogeneous PBW-algebra, i.e. a homogeneous graded algebra having a Poincaré-Birkhoff-Witt basis. We find sufficient conditions for a non-locally finite homogeneous PBW-algebra to be Koszul, which allows us to completely determine the cohomology of the universal Steenrod algebra at any prime.

Binomial Skew Polynomial Rings, Artin-Schelter Regularity, and Binomial Solutions of the Yang-Baxter Equation

Gateva-Ivanova, Tatiana (2004)

Serdica Mathematical Journal

2000 Mathematics Subject Classification: Primary 81R50, 16W50, 16S36, 16S37.Let k be a field and X be a set of n elements. We introduce and study a class of quadratic k-algebras called quantum binomial algebras. Our main result shows that such an algebra A defines a solution of the classical Yang-Baxter equation (YBE), if and only if its Koszul dual A! is Frobenius of dimension n, with a regular socle and for each x, y ∈ X an equality of the type xyy = αzzt, where α ∈ k {0, and z, t ∈ X is satisfied...

Ext-algebras and derived equivalences

Dag Madsen (2006)

Colloquium Mathematicae

Using derived categories, we develop an alternative approach to defining Koszulness for positively graded algebras where the degree zero part is not necessarily semisimple.

From factorizations of noncommutative polynomials to combinatorial topology

Vladimir Retakh (2010)

Open Mathematics

This is an extended version of a talk given by the author at the conference “Algebra and Topology in Interaction” on the occasion of the 70th Anniversary of D.B. Fuchs at UC Davis in September 2009. It is a brief survey of an area originated around 1995 by I. Gelfand and the speaker.

Hochschild cohomology of socle deformations of a class of Koszul self-injective algebras

Nicole Snashall, Rachel Taillefer (2010)

Colloquium Mathematicae

We consider the socle deformations arising from formal deformations of a class of Koszul self-injective special biserial algebras which occur in the study of the Drinfeld double of the generalized Taft algebras. We show, for these deformations, that the Hochschild cohomology ring modulo nilpotence is a finitely generated commutative algebra of Krull dimension 2.

Koszul and quasi-Koszul algebras obtained by tilting

R. M. Aquino, E. L. Green, E. N. Marcos (2002)

Colloquium Mathematicae

Given a finite-dimensional algebra, we present sufficient conditions on the projective presentation of the algebra modulo its radical for a tilted algebra to be a Koszul algebra and for the endomorphism ring of a tilting module to be a quasi-Koszul algebra. One condition we impose is that the algebra has global dimension no greater than 2. One of the main techniques is studying maps between the direct summands of the tilting module. Some applications are given. We also show that a Brenner-Butler...

Koszul duality and semisimplicity of Frobenius

Pramod N. Achar, Simon Riche (2013)

Annales de l’institut Fourier

A fundamental result of Beĭlinson–Ginzburg–Soergel states that on flag varieties and related spaces, a certain modified version of the category of -adic perverse sheaves exhibits a phenomenon known as Koszul duality. The modification essentially consists of discarding objects whose stalks carry a nonsemisimple action of Frobenius. In this paper, we prove that a number of common sheaf functors (various pull-backs and push-forwards) induce corresponding functors on the modified category or its triangulated...

Koszul duality for N-Koszul algebras

Roberto Martínez-Villa, Manuel Saorín (2005)

Colloquium Mathematicae

The correspondence between the category of modules over a graded algebra and the category of graded modules over its Yoneda algebra was studied in [8] by means of A algebras; this relation is very well understood for Koszul algebras (see for example [5],[6]). It is of interest to look for cases such that there exists a duality generalizing the Koszul situation. In this paper we will study N-Koszul algebras [1], [7], [9] for which such a duality exists.

Stratified modules over an extension algebra

Erzsébet Lukács, András Magyar (2018)

Czechoslovak Mathematical Journal

Let A be a standard Koszul standardly stratified algebra and X an A -module. The paper investigates conditions which imply that the module Ext A * ( X ) over the Yoneda extension algebra A * is filtered by standard modules. In particular, we prove that the Yoneda extension algebra of A is also standardly stratified. This is a generalization of similar results on quasi-hereditary and on graded standardly stratified algebras.

The Hochschild cohomology ring modulo nilpotence of a stacked monomial algebra

Edward L. Green, Nicole Snashall (2006)

Colloquium Mathematicae

This paper studies the Hochschild cohomology of finite-dimensional monomial algebras. If Λ = K/I with I an admissible monomial ideal, then we give sufficient conditions for the existence of an embedding of K [ x , . . . , x r ] / x a x b f o r a b into the Hochschild cohomology ring HH*(Λ). We also introduce stacked algebras, a new class of monomial algebras which includes Koszul and D-Koszul monomial algebras. If Λ is a stacked algebra, we prove that H H * ( Λ ) / K [ x , . . . , x r ] / x a x b f o r a b , where is the ideal in HH*(Λ) generated by the homogeneous nilpotent elements. In...

Un complexe de Koszul de modules instables et cohomotopie d’un spectre de Thom

Nguyen Dang Ho Hai (2012)

Bulletin de la Société Mathématique de France

Dans [8], les auteurs ont construit une résolution injective minimale d’un module instable dans la catégorie des modules instables modulo 2 . A partir de cette résolution, un résultat de type conjecture de Segal a été obtenu pour un certain spectre de Thom. Le but de cet article est de refaire ces résultats pour les premiers impairs. Etant donné un premier impair p , on construit dans ce travail un complexe de Koszul dans la catégorie des modules instables sur l’algèbre de Steenrod modulo p . Une résolution...

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