# Are perfect complexes the same as compact objects in D(R) for noncommutative rings?

The Stacks Project proves Thomason's insight that

compact objects of the derived category $$\simeq$$ bounded complexes of finitely generated projective modules

in Section 15.78, but the running conventions of the Stacks Project assume that all rings are commutative.

I believe the insight is originally from the Thomason-Trobaugh paper in the Grothendieck Festschrift, but this paper also works exclusively with commutative rings (and schemes).

What is the situation for non-commutative rings?

First, you can notice that if you have a set of compact objects $$F$$ in a triangulated category $$S$$ such that there's no proper triangulated, coproduct-closed subcategory containing $$F$$, the minimal thick subcategory containing $$F$$ is exactly the subcategory of compact objects. (It's more or less a consequence of the triangulated Brown representability theorem.)
Then if $$T$$ is $$D(R{\operatorname{-mod}})$$, $$R$$ is a compact object in it (since homs from it are zeroth cohomology); the smallest thick subcategory containing $$R$$ is the subcategory of perfect complexes, so by the observation above it is the subcategory of compact objects.