Eilenberg–Watts theorem
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In mathematics, specifically homological algebra, the Eilenberg–Watts theorem tells when a functor between the categories of modules is given by an application of a tensor product. Precisely, it says that a functor F : M o d R → M o d S {\displaystyle F:\mathbf {Mod} _{R}\to \mathbf {Mod} _{S}} is additive, is right-exact and preserves coproducts if and only if it is of the form F ≃ − ⊗ R F ( R ) {\displaystyle F\simeq -\otimes _{R}F(R)}.
For a proof, see
- Charles E. Watts, Intrinsic characterizations of some additive functors, Proc. Amer. Math. Soc. 11, 1960, 5–8.
- Samuel Eilenberg, Abstract description of some basic functors, J. Indian Math. Soc. (N.S.) 24, 1960, 231–234 (1961).