For hereditary complete cotorsion pairs generated by a set in a Grothendieck category, the coderived category of the left class is equivalent to the contraderived category of the right class, and for sandwiched pairs this flat-type behavior is equivalent to two periodicity properties.
Relative nonhomogeneous Koszul duality
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abstract
This book contains a detailed exposition of the nonhomogeneous Koszul duality theory in the relative situation over a noncentral, noncommutative, nonsemisimple base ring, as announced in Section 0.4 of arXiv:0708.3398. We prove the Poincare-Birkhoff-Witt theorem in this context and construct the triangulated equivalences of derived Koszul duality. The duality between the ring of differential operators and the de Rham DG-algebra, with the ring of functions as the base ring, is the thematic example. The moderate generality level makes the exposition in this book more accessible than the very heavily technical Chapter 11 of arXiv:0708.3398.
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Coderived and contraderived categories for a cotorsion pair, flat-type cotorsion pairs, and relative periodicity
For hereditary complete cotorsion pairs generated by a set in a Grothendieck category, the coderived category of the left class is equivalent to the contraderived category of the right class, and for sandwiched pairs this flat-type behavior is equivalent to two periodicity properties.