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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 110.17. 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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