@MISC{Keller93derivingdg, author = {Bernhard Keller}, title = {Deriving Dg Categories}, year = {1993} }

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Abstract

We investigate the (unbounded) derived category of a differential Z-graded category (=DG category). As a first application, we deduce a 'triangulated analogue` (4.3) of a theorem of Freyd's [5, Ex. 5.3 H] and Gabriel's [6, Ch. V] characterizing module categories among abelian categories. After adapting some homological algebra we go on to prove a 'Morita theorem` (8.2) generalizing results of [19] and [20]. Finally, we develop a formalism for Koszul duality [1] in the context of DG augmented categories. Summary We give an account of the contents of this paper for the special case of DG algebras. Let k be a commutative ring and A a DG (k-)algebra, i.e. a Z-graded k-algebra A = a p2Z A p endowed with a differential d of degree 1 such that d(ab) = (da)b + (\Gamma1) p a(db) for all a 2 A p , b 2 A. A DG (right) A-module is a Z-graded A-module M = ` p2Z M p endowed with a differential d of degree 1 such that d(ma) = (dm)a + (\Gamma1) p m(da) for all m 2 M p , a 2 A. A morphism of DG A-modules is a homogeneous morphism of degree 0 of the underlying graded A-modules commuting with the differentials. The DG A-modules form an abelian category CA. A morphism f : M ! N of CA is null-homotopic if f = dr + rd for some homogeneous morphism r : M ! N of degree-1 of the underlying graded A-modules.