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What are examples that show that derived categories really makes things easier/more transparent/have a real use?

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You may want to read some of the pointers given here. – t.b. Oct 31 '11 at 18:05
Do you like Morita equivalence? :-) It can be generalized in presence of differential structures, and even A-infty structures using derived categories. In general, derived categories are quite natural if you want to study structures with differentials (like complexes of modules over a given ring) identifying those structures which give the same information, i.e. cohomology. Localization w.r.t. quasi-isomorphisms is the method behind this idea. – Avitus Jun 11 '13 at 9:29

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