In a paper I recently read something about the "unibranch partial normalization" of a curve.

Say, $R$ is a local integral domain with maximal ideal $\mathfrak{m}$ and fraction field $K$. Is it possible to find a minimal ring $R^u$ between $R$ and its normalization $R'$ in $K$ such that all localizations of $R^u$ in the maximal ideals lying above $\mathfrak{m}$ are unibranch? Clearly, $R'$ itself satisfies all properties except for being minimal with respect to being unibranch "above" $\mathfrak{m}$.

I guess this is not what is considered to be the unibranch partial normalization in the paper above but I would still like to know if this is possible. One point I find interesting is the following: In general, if $R$ is noetherian, the normalization $R'$ is not noetherian. But is perhaps $R^u$ noetherian if $R$ is noetherian?

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    $\begingroup$ The key thing about curves (over a field, say) is that the normalization $\widetilde{R}$ of a local ring $R$ at a closed point is not only module-finite but the cokernel $R$-module $\widetilde{R}/R$ is of finite length (as curves have dimension 1) and so is unaffected by scalar extension to the henselization $R^{\rm{h}}$. That is why one can "pull down" irreducible components of the normalization of $R^{\rm{h}}$ to get an $R$-subalgebra of $\widetilde{R}$ to define a "unbranched partial normalization". It seems unrealistic to expect anything like that beyond dimension 1. $\endgroup$
    – grghxy
    Aug 31, 2015 at 14:32


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