Let $R$ be a commutative and reduced ring, finitely presented over $\mathbb Z$. Let $\overline R$ be the integral closure of $R$ in its total ring of fractions. In https://arxiv.org/abs/alg-geom/9704017, T. De Jong describes an algorithm to compute $\overline R$ explicitly ($R$ is of finite type, so it is excellent and De Jong's hypotheses hold). In summary, the algorithm is:
(1) Identify a certain radical ideal $I\subset R$ such that we have injective maps of $R$-algebras $R\subset \operatorname{Hom}_{R-Mod}(I,I) \subset \overline R$, with equality in the first inclusion if and only if $R=\overline R$.
(2) Replace $R$ by $\operatorname{Hom}_{R-Mod}(I,I)$ and repeat until $R$ is normal.
Suppose given a finite presentation for $R$ over $\mathbb Z$. I would like to end up with a finite presentation for $\overline R$. I understand how to obtain generators and relations for a suitable ideal $I$ satisfying (1), but not how to get a finite presentation for $\operatorname{Hom}_{R-Mod}(I,I)$. Writing $$ \operatorname{Hom}_{R-Mod}(I,I)=\operatorname{Hom}_{R-Mod}(I,R)\otimes_RI $$ reduces the problem to finding generators and relations for the dual module $I^\vee:=\operatorname{Hom}_{R-Mod}(I,R)$. We naturally have access to equations for $I^\vee$ inside a free $R$-module. To get a presentation, one needs to "solve these equations". How do we do it?
As far as I can tell, the algorithm is implemented in Macaulay2, but I have a hard time reading the documentation. Given two maps \begin{align*} f_A & \colon R^m \to R^n \\ f_B & \colon R^k \to R^n, \end{align*} Macaulay2 represents the module $I:=(\operatorname{Im}f_A + \operatorname{Im}f_B)/\operatorname{Im}f_B$ as a triple $(n,A,B)$ where $A,B$ are the matrices of $f_A,f_B$. Dualizing within this data structure is easy, but the missing piece of the puzzle is now: once we have represented $\operatorname{Hom}_{R-Mod}(I,I)$ as such a triple, how do we obtain a finite presentation for it as a $R$-algebra?
