# Questions about $\text{Perf}(A)$ of dg algebra $A$

[ALEXEY ELAGIN AND VALERY A. LUNTS, p.4.] Recall that triangulated category $$\text{Perf}(A)$$ is defined as the full thick triangulated subcategory of $$D(A)$$ generated by the dg $$A$$-module $$A$$.

[Kontsevich, Definition 1.] Dg algebra $$A$$ is called smooth if $$A \in \operatorname{Perf}\left(A \otimes A^{o p}\right)$$. It is compact if $$\operatorname{dim} H^{\bullet}(A, d)<\infty$$. This properties are preserved under the derived Morita equivalence.

Could you please share your understanding of $$\text{Perf}(A)$$?

• What is the relation between $$\text{Perf}(A)$$ and perfect complex?
• What good properties does it have?
• Why do we use $$\text{Perf}(A\otimes A^{op})$$ to define "smoothness"? What is the geometrical motivation?

Thank you very much!

As explained a little bit further in Elagin and Lunts' paper, the category $$Perf(A)$$ consists of the compact objects of $$D(A)$$, this is exactly what happens in the usual situation in algebraic geometry, for example for a nice space and certainly for a commutative ring, the perfect complexes as locally quasi-isomorphic to a bounded complex of free modules of finite type. The proof that these two notions are the same in D(R) can be found for example in the tag 07LT in the stacks project.
The so to say slogan of perfect complexes at least of good spaces is that they're finite, like the previous characterization as compact objects it is possible to show too that in a lot of situations of interest they are also the dualizable objects under the usual monoidal structure of $$D(A)$$. I won't recall what the definition of dualizable is ( but you can check the nlab entry ) but remember that finite dimensional vector spaces are what you must be thinking of. In this sense then perfect complexes are objects which are both topologically and algebraically finite. This means that perfect complexes are categorically closed under a lot operations you would expect f.d. vector spaces or compact (topological) spaces to be closed by.