# nLab perfect dg-modules

## Idea

The perfect dg-modules over a dg-category are the compact objects of its derived dg-category.

## 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

By the explicit description of the pretriangulated envelope, one gets

###### Lemma

A dg-module $M \in D(T)$ is perfect if and only if it is it is in the full sub-dg-category of $D(T)$ generated by the finitely generated semi-free dg-modules under direct summands.

###### 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

Created on January 7, 2015 at 12:11:16. See the history of this page for a list of all contributions to it.