# Dualities on Generalized Koszul Algebras

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*Edward L. Green; Idun Reiten; Øyvind Solberg*

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.

#### Table of Contents

# Table of Contents

## Dualities on Generalized Koszul Algebras

- Contents vii8 free
- Introduction ix10 free
- Acknowledgment xi12 free
- Notation and general preliminary results xiii14 free
- Chapter I. Main results and examples 118 free
- Chapter II. Proofs of main results 1734
- Chapter III. Generalized T-Koszul algebras 3047
- Chapter IV. Further results and questions 4865
- Bibliography 6683