The Stacks project

109.4 Tensor product

Tensor products are introduced in Algebra, Section 10.11. Let $R$ be a ring. Let $\text{Mod}_ R$ be the category of $R$-modules. We will say that a functor $F : \text{Mod}_ R \to \text{Mod}_ R$

  1. 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.

  2. $R$-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 $M, N$,

  3. 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,

  4. 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,

  5. 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 109.4.1. Let $R$ be a ring. With notation as above.

  1. Give an example of a ring $R$ and an additive functor $F : \text{Mod}_ R \to \text{Mod}_ R$ which is not $R$-linear.

  2. 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,

  3. 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 $N$.

  4. 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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