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Two Kinds of Derived Categories, Koszul Duality, and Comodule-Contramodule Correspondence
Leonid Positselski, Institute for Information Transmission Problems, Moscow, Russia

Memoirs of the American Mathematical Society
2011; 133 pp; softcover
Volume: 212
ISBN-10: 0-8218-5296-5
ISBN-13: 978-0-8218-5296-5
List Price: US$75
Individual Members: US$45
Institutional Members: US$60
Order Code: MEMO/212/996
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The aim of this paper is to construct the derived nonhomogeneous Koszul duality. The author considers 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.

Table of Contents

  • Introduction
  • Derived category of DG-modules
  • Derived categories of DG-comodules and DG-contramodules
  • Coderived and contraderived categories of CDG-modules
  • Coderived category of CDG-comodules and contraderived category of CDG-contramodules
  • Comodule-contramodule correspondence
  • Koszul duality: Conilpotent and nonconilpotent cases
  • \(\mathrm{A}_\infty\)-algebras and curved \(\mathrm{A}_\infty\)-coalgebras
  • Model categories of DG-modules, CDG-comodules, and CDG-contramodules
  • Model categories of DG-algebras and CDG-coalgebras
  • Appendix A. Homogeneous Koszul duality
  • Appendix B. \(\mathcal{D}\)-\(\Omega\) duality
  • Bibliography
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