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Two kinds of derived categories, Koszul duality, and comodule-contramodule correspondence

About this Title

Leonid Positselski, Sector of Algebra and Number Theory, Institute for Information Transmission Problems, Bolshoy Karetny per. 19 str. 1, Moscow 127994, Russia

Publication: Memoirs of the American Mathematical Society
Publication Year: 2011; Volume 212, Number 996
ISBNs: 978-0-8218-5296-5 (print); 978-1-4704-0613-4 (online)
Published electronically: November 19, 2010
Keywords: DG-algebra, CDG-coalgebra, curved A-infinity coalgebra, comodule, contramodule, derived category of the second kind, triangulated category, model category, D-module, Koszul duality
MSC: Primary 18E30, 18G10, 16T15, 16S37, 14F10; Secondary 18G55, 17B55, 16E65, 18G15, 58J10

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Table of Contents


  • Introduction
  • 1. Derived Category of DG-Modules
  • 2. Derived Categories of DG-Comodules and DG-Contramodules
  • 3. Coderived and Contraderived Categories of CDG-Modules
  • 4. Coderived Category of CDG-Comodules and Contraderived Category of CDG-Contramodules
  • 5. Comodule-Contramodule Correspondence
  • 6. Koszul Duality: Conilpotent and Nonconilpotent Cases
  • 7. $\mathrm {A}_\infty$-Algebras and Curved $\mathrm {A}_\infty$-Coalgebras
  • 8. Model Categories of DG-Modules, CDG-Comodules, and CDG-Contramodules
  • 9. Model Categories of DG-Algebras and CDG-Coalgebras
  • Appendix A. Homogeneous Koszul Duality
  • Appendix B. $\mathcal {D}$–$\Omega$ Duality


The aim of this paper is to construct the derived nonhomogeneous Koszul duality. We consider the derived categories of DG-modules, DG-comodules, and DG-contramodules, the coderived and contraderived categories of CDG-modules, the coderived category of CDG-comodules, and the contraderived category of CDG-contramodules. The equivalence between the latter two categories (the comodule-contramodule correspondence) is established. Nonhomogeneous Koszul duality or “triality” (an equivalence between exotic derived categories corresponding to Koszul dual (C)DG-algebra and CDG-coalgebra) is obtained in the conilpotent and nonconilpotent versions. Various A-infinity structures are considered, and a number of model category structures are described. Homogeneous Koszul duality and D-$\Omega$ duality are discussed in the appendices.

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