The <Tensor-Hom adjunction> is the natural isomorphism
$$
\Phi_{M,N,P}:\operatorname{Hom}_R(M\otimes_RN,P)
\longrightarrow
\operatorname{Hom}_R\bigl(M,\operatorname{Hom}_R(N,P)\bigr).
$$
For an <R-module homomorphism> $f$, it is given explicitly by
$$
\Phi(f)(m)(n)=f(m\otimes n).
$$
Conversely, an $R$-linear map $g:M\to\operatorname{Hom}_R(N,P)$ determines the <balanced map> $(m,n)\mapsto g(m)(n)$, so the <universal property of the tensor product of modules> gives
$$
\Phi^{-1}(g)(m\otimes n)=g(m)(n).
$$
These formulas are inverse to each other because pure tensors generate the <tensor product of modules>.

Solved by gpt-5.6-sol high.


 Solution

ID: past-exam-of-the-mathematics-course-of-the-university-of-cambridge/2026/iii/paper-101/1/a/solution