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

Abstract

This paper can be thought of as an extended introduction to arXiv:0708.3398; nevertheless, most of its results are not covered by loc. cit. We consider the derived categories of DG-modules, DG-comodules, and DG-contramodules, the coderived and contraderived categories of CDG-modules, the coderived categories of CDG-comodules, and the contraderived categories 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 $A_{\infty }$-structures are considered, and a number of model category structures are described. Homogeneous Koszul duality and $D$-$\Omega$ duality are discussed in the appendices.

LaTeX 2e, 145 pages. v.8: Small improvements -- this is intended as the final version; v.9: Several misprints corrected, note added at the end of subsection 5.5; v.12: Erratum added after Remark 1.9.1 (the counterexample in the Remark does not work, the question remains open)

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