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arXiv · 0905.2621

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

Leonid Positselski
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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.

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Leonid Positselski. 2009-05-17. Two kinds of derived categories, Koszul duality, and comodule-contramodule correspondence. https://doi.org/10.1090/s0065-9266-2010-00631-8

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