The Stacks project

Remark 10.119.6. 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 10.119.3 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 110.16. For an example in characteristic $p > 0$ see Example 10.119.5. 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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