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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