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Dualities on Generalized Koszul Algebras

Edward L. Green Virginia Polytechnic Institute and State University, Blacksburg, VA
Idun Reiten Norwegian University of Science and Technology, Trondheim, Norway
Øyvind Solberg Norwegian University of Science and Technology, Trondheim, Norway
Available Formats:
Electronic ISBN: 978-1-4704-0347-8
Product Code: MEMO/159/754.E
List Price: $56.00 MAA Member Price:$50.40
AMS Member Price: $33.60 Click above image for expanded view Dualities on Generalized Koszul Algebras Edward L. Green Virginia Polytechnic Institute and State University, Blacksburg, VA Idun Reiten Norwegian University of Science and Technology, Trondheim, Norway Øyvind Solberg Norwegian University of Science and Technology, Trondheim, Norway Available Formats:  Electronic ISBN: 978-1-4704-0347-8 Product Code: MEMO/159/754.E  List Price:$56.00 MAA Member Price: $50.40 AMS Member Price:$33.60
• Book Details

Memoirs of the American Mathematical Society
Volume: 1592002; 67 pp
MSC: 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.

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
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Volume: 1592002; 67 pp
MSC: 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.

• III. Generalized $T$-Koszul algebras