*Context:* I'm giving an informal seminar/reading group collection of talks on derived categories, following on from earlier talks giving the abstract definition. I am starting to talk about $\mathcal{D}^{b}(\mathrm{Rep}\ kQ)$, i.e. derived categories for representations of quivers, and am going on later to talk about $\mathcal{D}^{b}(\mathrm{Coh}\ X)$, i.e. coherent sheaves.


*Question:* I want to motivate studying $\mathcal{D}^{b}(\mathrm{Coh}\ X)$ by saying, analogously to the representation-theoretic setting, "derived equivalence is good because it's not too strong but it still detects key properties such as..." - and here, I need some help. **What properties of a variety are detected by the derived category of coherent sheaves?**


I'm aware of the Bondal-Orlov reconstruction theorem, which is obviously as good as one could hope for but doesn't always hold. But one expects some invariants to be picked up by the derived category, especially cohomological ones, but are there others too? Of course I'm deliberately being vague about what I mean by "variety" here, as I expect the answer to vary depending on what one asks for.


*Disclaimers:* IANAAG (as they say), as you'll have guessed from the question. I'm aware of [this question on MSE](https://math.stackexchange.com/questions/587540/what-is-a-concrete-example-of-why-one-wants-to-have-a-derived-category-in-alge) and [this one on MO](https://mathoverflow.net/questions/42463/derived-categories-of-coherent-sheaves-suggested-references) and a few other similar ones but I feel that none tackle my particular query - if I missed one that does, mea culpa. I also know of some survey-type articles on the topic by various authors, where one might have expected to find this question addressed, but I haven't seen it. I'd be very open to a good reference in lieu of an answer. I also don't see a community wiki button but surely this should be, if the powers that be could make it so, please.



Which properties of a variety are detected by its derived category of coherent sheaves?
One can see some initial answers in Huybrechts' book "Fourier-Mukai Transforms in Algebraic Geometry"


1. ([Huybrechts], Prop 4.1) If two smooth projective varieties are derived equivalent, then they have the same dimension.
2. ([Huybrechts], Prop 3.10) If $X$ is Noetherian, then $X$ is connected if and only if $D^b(Coh X)$ is indecomposable.


I heard about the first part of the following argument as a folklore, someone should correct me if I am wrong:


(EDIT: The following is WRONG. However, there is some interesting discussion in the comments to this answer.)


3. The compact objects in $D^b(Coh X)$ are precisely those quasi-isomorphic to bounded complex of locally free sheaves, so $X$ is regular if and only if every object in $D^b(Coh X)$ is compact.


