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Foxby equivalence, complete modules, and torsion modules. (English) Zbl 1010.16009
An equivalence theory for derived categories over differential graded algebras is developed using the adjoint functors and the classes of objects for which the values of unit and counit of adjunction are isomorphisms (Auslander and Bass classes). Some properties of the new equivalence are shown. It is proved that both classical Foxby equivalence (for Noetherian, local, commutative rings) and the Morita equivalence for complete modules and torsion modules, developed by Dwyer and Greenless, arise as special cases. Moreover, a new instance of this theory (Matlis equivalence) gives a characterization of Gorenstein rings.

MSC:
16E45 Differential graded algebras and applications (associative algebraic aspects)
18E30 Derived categories, triangulated categories (MSC2010)
16D90 Module categories in associative algebras
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