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

• 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.