I have a technical question on commutative algebra. I am not an expert in the subject, and I would like to know if there are "typical conditions" making the following possible.


Let $\varphi:R\to S$ be a surjective ring homomorphisms (rings are commutative with identity). Let $X$ be an $S$-module, and regard it as an $R$-module as well via $\varphi$. I am interested in recovering the $S$-dual $\text{Hom}\_S(X,S)$ of $X$ from the $R$-dual $\text{Hom}\_R(X,R)$, in a natural way. Do you know of a way to do so, possibly under suitable supplementary hypothesis?


A naive guess that comes to mind is to try and consider $\text{Hom}\_R(X,R)\otimes\_R S$. How is this module related to $\text{Hom}\_S(X,S)$?


I am interested in the very special case where $S$ (resp. $R$) is a order in a product $\prod K\_i$ (resp. $\prod K\_i\times\prod L\_j$) of finitely many number fields, and where $X$ is a torsion free, finitely generated $S$-module such that $X\otimes\mathbf{Q}$ is isomorphic to $\prod K\_i^{n\_i}$, for some $n\_i\geq 1$. Moreover both $S$ and $R$ are Gorenstein.



Behavior of duality under pull-back
To to do this, one just has to replace the $\text{Hom}$ by an $\text{Ext}$. Let me explain. You already assumed that both $R$ and $S$ are Gorenstein, so that makes things easier. 


Since $R$ and $S$ are Gorenstein, they are locally equidimensional, and so let us assume that $R$ is pure dimension $d$ and $S$ is pure dimension $e$ (by localizing if necessary). Then if $X$ is finitely generated,


$$
\begin{array}{rl}
& \text{Hom}\_S(X, S) \\
= & \text{Hom}\_S(X, \omega\_S) \\
= & h^{-e}(R\text{Hom}\_S(X, \omega\_S[e])) \\
= & h^{-e}(R\text{Hom}\_R(X, \omega\_R[d]) \\
= & h^{d-e}(R \text{Hom}\_R(X, \omega\_R) \\
= & \text{Ext}^{d-e}\_R(X, \omega\_R) \\
= & \text{Ext}^{d-e}\_R(X, R).
\end{array}
$$
The fact that $\omega\_R = R$ and $\omega\_S = S$ comes from the fact that $R$ and $S$ are Gorenstein (localize further if necessary). The real content is the third equality which is a basic case of Grothendieck duality (really, it's a derived version of something like Hom-tensor adjointness, plus some formulas for the dualizing module).


