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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
Dualities on Generalized Koszul Algebras
eBook 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
Dualities on Generalized Koszul Algebras
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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
eBook 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.

    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
  • Requests
     
     
    Review Copy – for publishers of book reviews
    Permission – for use of book, eBook, or Journal content
    Accessibility – to request an alternate format of an AMS title
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.

Readership

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
Review Copy – for publishers of book reviews
Permission – for use of book, eBook, or Journal content
Accessibility – to request an alternate format of an AMS title
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