Geodesic
Dg algebras with enough idempotents, their $\mathrm{dg}$ modules and their derived categories
Algebra and discrete mathematics, Tome 23 (2017) no. 1, pp. 62-137

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We develop the theory $\mathrm{dg}$ algebras with enough idempotents and their $\mathrm{dg}$ modules and show their equivalence with that of small $\mathrm{dg}$ categories and their $\mathrm{dg}$ modules. We introduce the concept of $\mathrm{dg}$ adjunction and show that the classical covariant tensor-Hom and contravariant Hom-Hom adjunctions of modules over associative unital algebras are extended as $\mathrm{dg}$ adjunctions between categories of $\mathrm{dg}$ bimodules. The corresponding adjunctions of the associated triangulated functors are studied, and we investigate when they are one-sided parts of bifunctors which are triangulated on both variables. We finally show that, for a $\mathrm{dg}$ algebra with enough idempotents, the perfect left and right derived categories are dual to each other.
Keywords: $\mathrm{dg}$ algebra, $\mathrm{dg}$ module, $\mathrm{dg}$ category, $\mathrm{dg}$ functor, $\mathrm{dg}$ adjunction, homotopy category, derived category, derived functor. 
@article{ADM_2017_23_1_a6,
     author = {Manuel Saor{\'\i}n},
     title = {Dg algebras with enough idempotents, their $\mathrm{dg}$ modules and their derived categories},
     journal = {Algebra and discrete mathematics},
     pages = {62--137},
     publisher = {mathdoc},
     volume = {23},
     number = {1},
     year = {2017},
     language = {en},
     url = {http://geodesic.mathdoc.fr/item/ADM_2017_23_1_a6/}
}
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Manuel Saorín. Dg algebras with enough idempotents, their $\mathrm{dg}$ modules and their derived categories. Algebra and discrete mathematics, Tome 23 (2017) no. 1, pp. 62-137. http://geodesic.mathdoc.fr/item/ADM_2017_23_1_a6/