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

TL;DR

This paper develops the theory of dg algebras with enough idempotents, establishing their equivalence with small dg categories, and explores dg adjunctions and dualities in derived categories.

Contribution

It introduces the concept of dg adjunctions, extends classical adjunctions to dg modules, and studies dualities in derived categories of dg algebras with enough idempotents.

Findings

Equivalence between dg algebras with enough idempotents and small dg categories.

Extension of classical adjunctions to dg bimodule categories.

Duality between perfect derived categories of dg algebras with enough idempotents.

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.