# nLab perfect dg-module

## Idea

Let $T$ be a dg-category and $D(T)$ its derived dg-category. Consider the dg-Yoneda embedding

$h : T \hookrightarrow D(T).$

The thick triangulated subcategory generated by its essential image is denoted $Perf(T) \subset D(T)$. It coincides with the full subcategory of compact objects of $D(T)$.

## Definition

###### Definition

A dg-module $M \in D(T)$ is perfect if it is in the full sub-dg-category generated by the pretriangulated envelope $tri(A)$ under direct summands.

We will write $perf(T) \subset D(T)$ for the full sub-dg-category of $D(T)$ spanned by perfect dg-modules. This is a pretriangulated sub-dg-category.

## Properties

###### Lemma

A dg-module $M \in D(T)$ is compact if and only if it is it is perfect.

## References

Section 2.3 of

Paragraph 3.5 of

Last revised on December 15, 2018 at 08:44:51. See the history of this page for a list of all contributions to it.