# Dualities on generalized Koszul algebras

### About this Title

**Edward L. Green**, **Idun Reiten** and **Øyvind Solberg**

Publication: Memoirs of the American Mathematical Society

Publication Year
2002: Volume 159, Number 754

ISBNs: 978-0-8218-2934-9 (print); 978-1-4704-0347-8 (online)

DOI: http://dx.doi.org/10.1090/memo/0754

MathSciNet review: 1921583

MSC (2000): Primary 16S37; Secondary 16E05, 16E30, 16W50

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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.

Readership

Graduate 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