Let $SpecA=X$ be an affine noetherian scheme. Let $QCoh(X)$ denote the derived (stable $\infty$-)category of quasi-coherent sheaves on $X$. There are the following special full subcategories spanned by objects of a given type (subcategories means here that they are closed under finite (co-)limits aka shits and cones).

**Perfect**- Complexes quasi-isomorphic to some bounded complex of flat finite modules.**Coherent**- Complexes with non-zero coherent cohomology in finitely many degrees.**Finite tor dimension**- Complexes quasi-isomorphic to some bounded complex of flat modules**Finite injective dimension**- Complexes quasi-isomorphic to some bounded complex of injective modules**Pseudo-coherent**- Complexes with coherent cohomology concentrated in a bounded above range.

There is a famous characterization of perfect complexes as the compact objects in $QCoh(X)$.

Question 1: Is there a similarly slick (stable $\infty$-)categorical characterization for 2, 3, 4, 5 above?Suggestions:

- Perhaps $\mathcal{M}$ is of finite tor dimension iff $\mathcal{M} \otimes_{\mathcal{O}_X} (-)$ commutes with arbitrary products
- Perhaps $\mathcal{M}$ is of finite injective dimension iff $\mathcal{Hom}_{\mathcal{O}_X}(-, \mathcal{M})$ sends products to coproducts