Abstract:
We develop the theory dg algebras with enough
idempotents and their dg modules and show their equivalence with
that of small dg categories and their dg modules. We introduce the
concept of dg adjunction and show that the classical covariant tensor-
Hom and contravariant Hom-Hom adjunctions of modules over
associative unital algebras are extended as dg adjunctions between
categories of 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 dg algebra with enough
idempotents, the perfect left and right derived categories are dual
to each other.
