111.4 Tensor product
Tensor products are introduced in Algebra, Section 10.12. Let R be a ring. Let \text{Mod}_ R be the category of R-modules. We will say that a functor F : \text{Mod}_ R \to \text{Mod}_ R
is additive if F : \mathop{\mathrm{Hom}}\nolimits _ R(M, N) \to \mathop{\mathrm{Hom}}\nolimits _ R(F(M), F(N)) is a homomorphism of abelian groups for any R-modules M, N, see Homology, Definition 12.3.1.
R-linear if F : \mathop{\mathrm{Hom}}\nolimits _ R(M, N) \to \mathop{\mathrm{Hom}}\nolimits _ R(F(M), F(N)) is R-linear for any R-modules M, N,
right exact if for any short exact sequence 0 \to M_1 \to M_2 \to M_3 \to 0 the sequence F(M_1) \to F(M_2) \to F(M_3) \to 0 is exact,
left exact if for any short exact sequence 0 \to M_1 \to M_2 \to M_3 \to 0 the sequence 0 \to F(M_1) \to F(M_2) \to F(M_3) is exact,
commutes with direct sums, if given a set I and R-modules M_ i the maps F(M_ i) \to F(\bigoplus M_ i) induce an isomorphism \bigoplus F(M_ i) = F(\bigoplus M_ i).
Exercise 111.4.1. Let R be a ring. With notation as above.
Give an example of a ring R and an additive functor F : \text{Mod}_ R \to \text{Mod}_ R which is not R-linear.
Let N be an R-module. Show that the functor F(M) = M \otimes _ R N is R-linear, right exact, and commutes with direct sums,
Conversely, show that any functor F : \text{Mod}_ R \to \text{Mod}_ R which is R-linear, right exact, and commutes with direct sums is of the form F(M) = M \otimes _ R N for some R-module N.
Show that if in (3) we drop the assumption that F commutes with direct sums, then the conclusion no longer holds.
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