两类导出范畴、Koszul对偶和余模-反模对应
Two kinds of derived categories, Koszul duality, and comodule-contramodule correspondence
摘要：
这篇论文可以被认为是对arXiv：0708.3398的扩展介绍;然而，它的大部分结果都没有被引用。前引书我们考虑了DG-模、DG-余模和DG-逆模的导出范畴，CD-模的上导出范畴和逆导出范畴，CD-余模的上导出范畴，以及CD-逆模的逆导出范畴.后两个范畴之间的等价性（余模-反模对应）被建立。在幂上和非幂上两种形式下，得到了非齐次Koszul对偶或“triality”（对应于Koszul对偶（C）DG-代数和CD G-余代数的奇异导出范畴之间的等价性）.各种$A_\infty$ -结构被认为是，并描述了一些模型类别结构。齐次Koszul对偶和$D$ - $Ω$对偶在附录中讨论。
英文摘要：
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 $D$-$Ω$ duality are discussed in the appendices.
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