Tensor-hom adjunction

Remarks

The inverse sends $\psi \colon M \to \Hom_R(N, P)$ to the linear map $M \otimes_R N \to P$ induced by the bilinear map $(m, n) \mapsto \psi(m)(n)$. This is exactly the universal property of the tensor product: linear maps out of $M \otimes_R N$ are bilinear maps on $M \times N$. In turn, a bilinear map is a linear map from $M$ to linear maps on $N$.

The adjunction has consequences for exact sequences. The functor $\Hom_R(N, -)$ is left exact: if $0 \to A \to B \to C$ is exact, then so is $0 \to \Hom_R(N, A) \to \Hom_R(N, B) \to \Hom_R(N, C)$. As a left adjoint, $- \otimes_R N$ is right exact: if $A \to B \to C \to 0$ is exact, then so is $A \otimes_R N \to B \otimes_R N \to C \otimes_R N \to 0$. Neither functor is exact in general. For $N = \Z/2\Z$ over $\Z$, the surjection $\Z \to \Z/2\Z$ gives $\Hom(\Z/2\Z, \Z) = 0 \to \Hom(\Z/2\Z, \Z/2\Z) \cong \Z/2\Z$, which is not surjective; and the injection $\Z \xrightarrow{2} \Z$ becomes the zero map after $- \otimes_\Z \Z/2\Z$.

Going deeper

For a student

This is the prototype of an adjunction, and it explains at once why tensor products are right exact and $\Hom$ is left exact. The failure of exactness is measured by the derived functors $\operatorname{Tor}$ and $\operatorname{Ext}$. Flat modules are those $N$ for which $- \otimes_R N$ is exact, and projective modules those for which $\Hom_R(N, -)$ is exact. The same adjunction holds for bimodules over noncommutative rings, and for sheaves of modules.