nLab
perfect dg-module

Idea

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

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

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

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

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

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