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

• Dmitri Orlov, Smooth and proper noncommutative schemes and gluing of DG categories, arXiv:1402.7364.

Paragraph 3.5 of

• Bernhard Keller, On differential graded categories, arXiv:math/0601185.