For an additive right-exact functor on module categories, apply it to a projective resolution and take homology. Lifts of module maps exist by projectivity; any two lifts are chain homotopic, giving independence and functoriality. The zeroth derived functor is naturally , positive degrees vanish on projective modules, and the horseshoe lemma produces a long exact homology sequence from each short exact sequence. A Tor functor is the left derived functor of a tensor product of modules functor.
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