Electronic ISBN:  9781470403478 
Product Code:  MEMO/159/754.E 
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Book DetailsMemoirs of the American Mathematical SocietyVolume: 159; 2002; 67 ppMSC: Primary 16; Secondary 18;
Koszul rings are graded rings which have played an important role in algebraic topology, algebraic geometry, noncommutative algebraic geometry, and in the theory of quantum groups. One aspect of the theory is to compare the module theory for a Koszul ring and its Koszul dual. There are dualities between subcategories of graded modules; the Koszul modules. When \(\Lambda\) is an artin algebra and \(T\) is a cotilting \(\Lambda\)module, the functor \(\mathrm{Hom}_\Lambda(\ ,T)\) induces a duality between certain subcategories of the finitely generated modules over \(\Lambda\) and \(\mathrm{End}_\Lambda(T)\).
The purpose of this paper is to develop a unified approach to both the Koszul duality and the duality for cotilting modules. This theory specializes to these two cases and also contains interesting new examples. The starting point for the theory is a positively \(\mathbb{Z}\)graded ring \(\Lambda=\Lambda_0+\Lambda_1+\Lambda_2+\cdots\) and a (Wakamatsu) cotilting \(\Lambda_0\)module \(T\), satisfying additional assumptions. The theory gives a duality between certain subcategories of the finitely generated graded modules generated in degree zero over \(\Lambda\) on one hand and over the Yoneda algebra \(\oplus_{i\geq 0} \mathrm{Ext}^i_\Lambda(T,T)\) on the other hand.ReadershipGraduate students and research mathematicians interested in associative rings and algebras.

Table of Contents

Chapters

I. Main results and examples

II. Proofs of main results

III. Generalized $T$Koszul algebras

IV. Further results and questions


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Koszul rings are graded rings which have played an important role in algebraic topology, algebraic geometry, noncommutative algebraic geometry, and in the theory of quantum groups. One aspect of the theory is to compare the module theory for a Koszul ring and its Koszul dual. There are dualities between subcategories of graded modules; the Koszul modules. When \(\Lambda\) is an artin algebra and \(T\) is a cotilting \(\Lambda\)module, the functor \(\mathrm{Hom}_\Lambda(\ ,T)\) induces a duality between certain subcategories of the finitely generated modules over \(\Lambda\) and \(\mathrm{End}_\Lambda(T)\).
The purpose of this paper is to develop a unified approach to both the Koszul duality and the duality for cotilting modules. This theory specializes to these two cases and also contains interesting new examples. The starting point for the theory is a positively \(\mathbb{Z}\)graded ring \(\Lambda=\Lambda_0+\Lambda_1+\Lambda_2+\cdots\) and a (Wakamatsu) cotilting \(\Lambda_0\)module \(T\), satisfying additional assumptions. The theory gives a duality between certain subcategories of the finitely generated graded modules generated in degree zero over \(\Lambda\) on one hand and over the Yoneda algebra \(\oplus_{i\geq 0} \mathrm{Ext}^i_\Lambda(T,T)\) on the other hand.
Graduate students and research mathematicians interested in associative rings and algebras.

Chapters

I. Main results and examples

II. Proofs of main results

III. Generalized $T$Koszul algebras

IV. Further results and questions