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Any abelian group can be reconstructed from its rationalized and p-completed parts. This object-wise decomposition can be lifted to several categorical decompositions of D(Z) -- the derived category of abelian groups. In fact, given any (suitably well-behaved) commutative ring R, a similar process can be performed where D(R) is decomposed using the data of its Zariski prime spectrum.
Links:
[1] https://www.mpim-bonn.mpg.de/de/taxonomy/term/39
[2] https://www.mpim-bonn.mpg.de/de/TopologySeminar