Tensor–Hom adjunction

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

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