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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 and this one on MO 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.

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    $\begingroup$ Sort of aside maybe - there seems to be lot of additional structure, simplest one being derived tensor product. I wonder whether there are theorems describing all of the structure involved. $\endgroup$ – მამუკა ჯიბლაძე Jan 12 '15 at 12:46
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    $\begingroup$ There are other ways to motivate the derived category than touting it as a good invariant, an important one being defining derived functors on them. $\endgroup$ – Qiaochu Yuan Jan 12 '15 at 13:02
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    $\begingroup$ @QiaochuYuan Certainly, I agree. But I wanted to have a synergy between my stories for the representation theory and for the algebraic geometry, so it seemed a natural question to ask. $\endgroup$ – Jan Grabowski Jan 12 '15 at 13:56
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    $\begingroup$ @მამუკაჯიბლაძე, when taking into account the derived tensor product, one can recover the variety completely (this is a theorem of Thomason and Balmer). $\endgroup$ – AAK Jan 12 '15 at 14:39
  • $\begingroup$ Made CW on OP's request. $\endgroup$ – Todd Trimble Jan 12 '15 at 17:21
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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.)

  1. 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.
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  • $\begingroup$ 3. is slightly different: $X$ is regular iff $D^b(Coh(X))\subset D(QCoh(X))$ is the subcategory of compact objects. $\endgroup$ – Fernando Muro Jan 12 '15 at 13:52
  • $\begingroup$ Thanks @waikit - they are definitely along the lines I had in mind. $\endgroup$ – Jan Grabowski Jan 12 '15 at 13:57
  • $\begingroup$ Thanks for the correction. Can the argument be fixed by reconstructing $D(QCoh(X))$ from $D^b(Coh(X))$? Perhaps by defining it as the collection of formal cones of direct sums of object in $D^b(Coh(X))$? $\endgroup$ – Wai-kit Yeung Jan 12 '15 at 14:01
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    $\begingroup$ As far as I know, it is not possible to recover the derived category of quasi-coherent complexes from the bounded derived category of coherent sheaves, at the triangulated level. At the level of infinity- or dg-categories, one can recover it as the ind-objects (in the regular case). $\endgroup$ – AAK Jan 12 '15 at 14:31
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    $\begingroup$ @waikit as Adeel says, although at the higher categorical level $D(QCoh(X))$ is the free cocompletion of $D^b(Coh(X))$ at the triangulated level there's no notion of cocompletion. However there are still open questions in this direction, like the Margolis Conjecture, which is the analogue of this in stable homotopy theory mathoverflow.net/questions/67227/… $\endgroup$ – Fernando Muro Jan 16 '15 at 11:29

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