Embeddings of derived categories of bornological modules

Let A be an algebra with a countable basis and let B be, say, a Frechet algebra that contains A as a dense subalgebra. This embedding induces a functor from the derived category of B-modules to the derived category of A-modules. In many important examples, this functor is fully faithful. We study this property in some detail, giving several equivalent conditions, examples, and applications. To prepare for this, we explain carefully how to do homological algebra with modules over bornological algebras. We construct the derived category of bornological left A-modules and some standard derived functors, with special emphasis on the adjoint associativity between the tensor product and the internal Hom functor. We also discuss the category of essential modules over a non-unital algebra and its functoriality.