Embeddings of derived categories of bornological modules

TL;DR

This paper investigates the conditions under which the embedding of a dense subalgebra into a larger algebra induces a fully faithful functor between their derived categories, with applications to bornological modules.

Contribution

It provides a detailed analysis of the fully faithfulness of the embedding-induced functor, including new conditions, examples, and applications in the context of bornological modules.

Findings

The functor is fully faithful in many important cases.

Equivalent conditions for full faithfulness are established.

Applications to homological algebra over bornological algebras are demonstrated.

Abstract

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.