Memoirs of the American Mathematical Society 2011; 133 pp; softcover Volume: 212 ISBN10: 0821852965 ISBN13: 9780821852965 List Price: US$75 Individual Members: US$45 Institutional Members: US$60 Order Code: MEMO/212/996
 The aim of this paper is to construct the derived nonhomogeneous Koszul duality. The author considers the derived categories of DGmodules, DGcomodules, and DGcontramodules, the coderived and contraderived categories of CDGmodules, the coderived category of CDGcomodules, and the contraderived category of CDGcontramodules. The equivalence between the latter two categories (the comodulecontramodule correspondence) is established. Nonhomogeneous Koszul duality or "triality" (an equivalence between exotic derived categories corresponding to Koszul dual (C)DGalgebra and CDGcoalgebra) is obtained in the conilpotent and nonconilpotent versions. Various Ainfinity 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 DGmodules
 Derived categories of DGcomodules and DGcontramodules
 Coderived and contraderived categories of CDGmodules
 Coderived category of CDGcomodules and contraderived category of CDGcontramodules
 Comodulecontramodule correspondence
 Koszul duality: Conilpotent and nonconilpotent cases
 \(\mathrm{A}_\infty\)algebras and curved \(\mathrm{A}_\infty\)coalgebras
 Model categories of DGmodules, CDGcomodules, and CDGcontramodules
 Model categories of DGalgebras and CDGcoalgebras
 Appendix A. Homogeneous Koszul duality
 Appendix B. \(\mathcal{D}\)\(\Omega\) duality
 Bibliography
