The Stacks project

\begin{equation*} \DeclareMathOperator\Coim{Coim} \DeclareMathOperator\Coker{Coker} \DeclareMathOperator\Ext{Ext} \DeclareMathOperator\Hom{Hom} \DeclareMathOperator\Im{Im} \DeclareMathOperator\Ker{Ker} \DeclareMathOperator\Mor{Mor} \DeclareMathOperator\Ob{Ob} \DeclareMathOperator\Sh{Sh} \DeclareMathOperator\SheafExt{\mathcal{E}\mathit{xt}} \DeclareMathOperator\SheafHom{\mathcal{H}\mathit{om}} \DeclareMathOperator\Spec{Spec} \newcommand\colim{\mathop{\mathrm{colim}}\nolimits} \newcommand\lim{\mathop{\mathrm{lim}}\nolimits} \newcommand\Qcoh{\mathit{Qcoh}} \newcommand\Sch{\mathit{Sch}} \newcommand\QCohstack{\mathcal{QC}\!\mathit{oh}} \newcommand\Cohstack{\mathcal{C}\!\mathit{oh}} \newcommand\Spacesstack{\mathcal{S}\!\mathit{paces}} \newcommand\Quotfunctor{\mathrm{Quot}} \newcommand\Hilbfunctor{\mathrm{Hilb}} \newcommand\Curvesstack{\mathcal{C}\!\mathit{urves}} \newcommand\Polarizedstack{\mathcal{P}\!\mathit{olarized}} \newcommand\Complexesstack{\mathcal{C}\!\mathit{omplexes}} \newcommand\Pic{\mathop{\mathrm{Pic}}\nolimits} \newcommand\Picardstack{\mathcal{P}\!\mathit{ic}} \newcommand\Picardfunctor{\mathrm{Pic}} \newcommand\Deformationcategory{\mathcal{D}\!\mathit{ef}} \end{equation*}

Remark 10.118.6. Suppose that $R$ is a $1$-dimensional semi-local Noetherian domain. If there is a maximal ideal $\mathfrak m \subset R$ such that $R_{\mathfrak m}$ is not regular, then we may apply Lemma 10.118.3 to $(R, \mathfrak m)$ to get a finite ring extension $R \subset R_1$. (For example one can do this so that $\mathop{\mathrm{Spec}}(R_1) \to \mathop{\mathrm{Spec}}(R)$ is the blowup of $\mathop{\mathrm{Spec}}(R)$ in the ideal $\mathfrak m$.) Of course $R_1$ is a $1$-dimensional semi-local Noetherian domain with the same fraction field as $R$. If $R_1$ is not a regular semi-local ring, then we may repeat the construction to get $R_1 \subset R_2$. Thus we get a sequence

\[ R \subset R_1 \subset R_2 \subset R_3 \subset \ldots \]

of finite ring extensions which may stop if $R_ n$ is regular for some $n$. Resolution of singularities would be the claim that eventually $R_ n$ is indeed regular. In reality this is not the case. Namely, there exists a characteristic $0$ Noetherian local domain $A$ of dimension $1$ whose completion is nonreduced, see [Proposition 3.1, Ferrand-Raynaud] or our Examples, Section 102.15. For an example in characteristic $p > 0$ see Example 10.118.5. Since the construction of blowing up commutes with completion it is easy to see the sequence never stabilizes. See [Bennett] for a discussion (mostly in positive characteristic). On the other hand, if the completion of $R$ in all of its maximal ideals is reduced, then the procedure stops (insert future reference here).


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