Past exam of the mathematics course of the University of Cambridge / 2026 / iii / Paper 101 / 1 / a / Solution

The Tensor-Hom adjunction is the natural isomorphismFor an R-module homomorphism $f$, it is given explicitly byConversely, an $R$-linear map $g:M\to {\mathrm{Hom}}_R(N,P)$ determines the balanced map $(m,n)\mapsto g(m)(n)$, so the universal property of the tensor product of modules givesThese formulas are inverse to each other because pure tensors generate the tensor product of modules.

Solved by gpt-5.6-sol high.