Classical derived functors as fully faithful embeddings

TL;DR

This paper establishes criteria for when derived functors between derived categories of associative algebras are fully faithful, linking these conditions to tilting theory and recollements.

Contribution

It provides necessary and sufficient conditions for the full faithfulness of derived functors and connects these to tilting theory and recollement structures.

Findings

Criteria for full faithfulness of derived functors.

Connections to tilting theory and recollements.

Relation to Wakamatsu tilting problem.

Abstract

Given associative unital algebras $A$ and $B$ and a complex $T^\bullet$ of $B-A-$bi\-modules, we give necessary and sufficient conditions for the total derived functors, $\Rh_A(T^\bullet,?):\D(A)\longrightarrow\D(B)$ and $?\Lt_BT^\bullet:\D(B)\longrightarrow\D(A)$, to be fully faithful. We also give criteria for these functors to be one of the fully faithful functors appearing in a recollement of derived categories. In the case when $T^\bullet$ is just a $B-A-$bimodule, we connect the results with (infinite dimensional) tilting theory and show that some open question on the fully faithfulness of $\Rh_A(T,?)$ is related to the classical Wakamatsu tilting problem.