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

We will write perf(T)D(T)perf(T) \subset D(T) for the full sub-dg-category of D(T)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 MD(T)M \in D(T) is perfect if and only if it is it is in the full sub-dg-category of D(T)D(T) generated by the finitely generated semi-free dg-modules under direct summands.

Lemma

A dg-module MD(T)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.