Given a map $f:A\rightarrow B$ of rings, we get a functor $Res_f:\text{B-Mod}\rightarrow \text{A-Mod}$, induced by our map $f$. This has left adjoint $B\otimes_A \text{_}$ where the $B$ module structure is given by multiplication on the first factor. We also have a right adjoint $Hom_A(B,\text{_})$, where we premultiply to give the $B$ module structure.
If $B/A$ is free, then these functors are isomorphic, though it seems we need to pick a basis of $B/A$ to exhibit this.
Are these functors still isomorphic if $B$ is locally free over $A$?
Really, this is trying to understand the difference between the left and right adjoints of $f_\ast$ for a map of schemes, but I am more comfortable in the commutative algebra/modules setting, hence phrasing in this manner.