Two kinds of derived categories, Koszul duality, and comodule-contramodule correspondence
Two kinds of derived categories, Koszul duality, and comodule-contramodule correspondence
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DOI：
10.1090/s0065-9266-2010-00631-8
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发表时间：
2009-05
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通讯作者：
L. Positselski
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作者：
L. Positselski
本文可被视为对arXiv:0708.3398的一篇扩展引言；然而，其大部分结果并未被上述文献涵盖。我们考虑DG -模、DG -余模和DG -反模的导出范畴，CDG -模的上导出范畴和反导出范畴，CDG -余模的上导出范畴以及CDG -反模的反导出范畴。后两类范畴之间的等价性（余模 - 反模对应）得以建立。得到了非齐次Koszul对偶性或“三元性”（对应于Koszul对偶的(C)DG -代数和CDG -余代数的奇异导出范畴之间的等价性）的幂零和非幂零版本。考虑了各种$A_\infty$ -结构，并描述了一些模型范畴结构。在附录中讨论了齐次Koszul对偶性和$D$-$\Omega$对偶性。
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.