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We develop the theory dg algebras with enough idempotents and their dgmodules and show their equivalence with that of small dg categories and theirdg modules. We introduce the concept of dg adjunction and show that theclassical covariant tensor-Hom and contravariant Hom-Hom adjunctions of modulesover associative unital algebras are extended as dg adjunctions betweencategories of dg bimodules. The corresponding adjunctions of the associatedtriangulated functors are studied, and we investigate when they are one-sidedparts of bifunctors which are triangulated on both variables. We finally showthat, for a dg algebra with enough idempotents, the perfect left and rightderived categories are dual to each other.

Dg algebras with enough idempotents, their dg modules and their derived categories