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It is well known that the derived category of coherent sheaves (unbounded, bounded, and all cousins) on a scheme $X$ contain most - if not all (depending on specifics) - of the cohomological information about $X$.

I know there is no formal statement to this effect, but how inaccurate is to to think of the derived categories living over X as a kind of homotopy type of $X$?

Is there any hope of a "decomposition" of $X$ into a system $(X_{i})$ (say by iterated blowups) in such a way that the corresponding system of derived categories is an analogue of the Postnikov tower?

I apologise for how vague these questions are, I am just trying to centre myself in the theory.

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    $\begingroup$ Small comment: if $X$ is a variety over $\mathbf C$, we can also view it as topological space $X(\mathbf C)$. The quasicoherent derived categories (or even the isomorphism type of $X$ as an abstract scheme!) do not know everything about the classical homotopy type. For example, Serre showed that for a smooth projective variety $X$ over $\mathbf C$ and an abstract field automorphism $\sigma$ of $\mathbf C$, the spaces $X(\mathbf C)$ and $^\sigma\!X(\mathbf C)$ can have different fundamental groups. $\endgroup$ Apr 12 '20 at 19:44
  • $\begingroup$ Maybe you're not asking about the comparison with the classical homotopy type, but if not then I'm not sure what you mean by 'a kind of homotopy type of $X$'. $\endgroup$ Apr 12 '20 at 19:52
  • $\begingroup$ True I have not been very specific. I guess what I mean by "homotopy type" ought to be a minimal collection of data encoding the information accessible by cohomological methods. Another essential piece is that this data be amenable to some sort of truncation or variation operation in the way the spaces have Postnikov decompositions. $\endgroup$ Apr 13 '20 at 12:49
  • $\begingroup$ I suppose I am picturing a world in which increasingly complex singularities are analogous to higher homtopical data. In this way the chain of blowups in a resolution of singularities (in characteristic 0) would be something like a Postnikov decomposition in reverse. $\endgroup$ Apr 13 '20 at 12:51
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    $\begingroup$ Informally, both derived categories and (stable) homotopy types provide a functor from a nonlinear category of spaces to a more linear category. Both formalisms relate to Grothendieck's idea of motives which is supposed to be a universal such functor. Indeed there are conjectural links between motives and derived categories, as well as adjoint functors between triangulated category of motives and A^1-stable homotopy category. Finally, one analogy for Postnikov towers in derived category is the concept of a semiorthognal decomposition. $\endgroup$ Apr 14 '20 at 21:25

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