Tensor–Hom adjunction

The tensor–Hom adjunction (over a commutative ring $R$ ) is the natural isomorphism of abelian groups (indeed $R$ -modules)

\mathrm{Hom}_R(M\otimes_R N,\,P)\;\cong\;\mathrm{Hom}_R\!\bigl(M,\,\mathrm{Hom}_R(N,P)\bigr),

natural in $M,N,P$ , where $\otimes_R$ is the tensor product and $\mathrm{Hom}_R(N,P)$ is the Hom module.

Remarks

Concretely, a map $f\colon M\otimes_R N\to P$ corresponds to the map $\Phi(f)\colon M\to \mathrm{Hom}_R(N,P)$ defined by $\Phi(f)(m)(n)=f(m\otimes n)$ , with inverse $\Psi(g)(m\otimes n)=g(m)(n)$ . This adjunction is a formal consequence of the universal property of the tensor product: it linearizes bilinear maps, and it specializes to duality when $P=R$ , giving $\mathrm{Hom}_R(M\otimes_R N,R)\cong \mathrm{Hom}_R(M,N^\vee)$ with $N^\vee$ the dual module

Examples

Bilinear pairings $M\times N\to P$ correspond to linear maps $M\to \mathrm{Hom}_R(N,P)$ .
Taking $P=R$ identifies bilinear forms $M\times N\to R$ with linear maps $M\to N^\vee$ .