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

Question

[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!

Answer 1

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.

These two nice properties are so that a lot of the geometry of the would-be space is reflected as properties of the category. As explained a bit by Kontsevich in the paper you list, a fundamental result is that of Bondal and Van der Bergh relating the category of perfect complexes of a separated scheme of finite type X with the category of perfect complexes of a single dg-algebra A. In particular if X is also smooth then the dg-algebra will be smooth in the sense of your question. This sort of characterization allows people to work purely axiomatically.

I hope this helps clarify some things and I apologize if my answer seems too vague. Perfect complexes appear in many ways and its a bit hard to list every good property and interesting result without knowing exactly from which angle the other person might be coming from.