Let $R$ be a Noetherian domain which is non-negatively graded over a field $K$, and is a finitely generated $K$-algebra, and let $\overline{R}$ be its normalization in the field of fractions of $R$. Further, suppose that the inclusion map $R \rightarrow \overline{R}$ is finite, i.e. that $\overline{R}$ is a finitely generated module over $R$ (the graded hypothesis may make this assumption redundant). I am interested in knowing how generally I can show that the cokernel of the natural map above is of finite length as an $R$-module.
In general, is there some easy way of knowing the dimension of the cokernel $\overline{R}/R$ viewed as an $R$-module, say just from a minimal presentation of $R$ as $K[x_1,...,x_n]/I$? Of course, the conductor ideal $\mathfrak{c}$ plays an important role, as the support of $\overline{R}/R$ is clearly the vanishing set of $\mathfrak{c}$, i.e. we have an equality: $\operatorname{Supp}_R(\overline{R}/R) = V(\mathfrak{c}) = \{ \mathfrak{p} \in \operatorname{Spec}(R) \mid \mathfrak{p} \supset \mathfrak{c}\}$.
But, this just pushes the question one step backwards -- when is the conductor ideal primary to the homogeneous maximal ideal of $R$? In general, is the height of $\mathfrak{c}$ easy to compute from a presentation as above? I have Macaulay2 code to compute it for specific examples, but I would like a reference for the general case outlined above. I also specifically want to avoid results which only consider the cases below.
- $K$ is of characteristic $0$ (I am interested primarily in the case $K$ is a perfect field of prime characteristic)
- $R$ is Cohen-Macaulay
- $R$ is of small dimension ($\dim(R)\le 1$)