Hom–tensor adjunction

For a commutative ring \(R\) and \(R\)-modules \(M,N,P\), there is a natural Hom–tensor adjunction \[ \operatorname{Hom}_R(M\otimes_R N,P) \cong \operatorname{Hom}_R\!\left(M,\operatorname{Hom}_R(N,P)\right). \] The isomorphism sends \(F:M\otimes_RN\to P\) to the map \[ m\longmapsto(n\mapsto F(m\otimes n)). \] For noncommutative rings the same statement requires the usual left/right module and bimodule hypotheses.

In representation theory, the corresponding induction–restriction adjunction is Frobenius reciprocity. For \(H\le G\), \[ \operatorname{Hom}_{\mathbb C[G]} \bigl(\mathbb C[G]\otimes_{\mathbb C[H]}V,W\bigr) \cong \operatorname{Hom}_{\mathbb C[H]} \bigl(V,\operatorname{Res}_H^GW\bigr). \] Since \[ \operatorname{Ind}_H^GV =\mathbb C[G]\otimes_{\mathbb C[H]}V, \] this is usually written \[ \operatorname{Hom}_G(\operatorname{Ind}_H^GV,W) \cong \operatorname{Hom}_H(V,\operatorname{Res}_H^GW). \]