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ISSN 1947-6221(e) ISSN 0065-9266(p)

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

Author(s): Leonid Positselski
Journal: Memoirs of the AMS 212 (2011), no. 996.
MSC (2010): Primary 18E30, 18G10, 16T15, 16S37, 14F10; Secondary 18G55, 17B55, 16E65, 18G15, 58J10
Posted: November 19, 2010
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Abstract: 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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