Loading web-font TeX/Main/Regular

The Stacks project

Lemma 54.13.2. Let (A, \mathfrak m) be a local Noetherian ring. Let B \subset C be finite A-algebras. Assume that (a) B is a normal ring, and (b) the \mathfrak m-adic completion C^\wedge is a normal ring. Then B^\wedge is a normal ring.

Proof. Consider the commutative diagram

\xymatrix{ B \ar[r] \ar[d] & C \ar[d] \\ B^\wedge \ar[r] & C^\wedge }

Recall that \mathfrak m-adic completion on the category of finite A-modules is exact because it is given by tensoring with the flat A-algebra A^\wedge (Algebra, Lemma 10.97.2). We will use Serre's criterion (Algebra, Lemma 10.157.4) to prove that the Noetherian ring B^\wedge is normal. Let \mathfrak q \subset B^\wedge be a prime lying over \mathfrak p \subset B. If \dim (B_\mathfrak p) \geq 2, then \text{depth}(B_\mathfrak p) \geq 2 and since B_\mathfrak p \to B^\wedge _\mathfrak q is flat we find that \text{depth}(B^\wedge _\mathfrak q) \geq 2 (Algebra, Lemma 10.163.2). If \dim (B_\mathfrak p) \leq 1, then B_\mathfrak p is either a discrete valuation ring or a field. In that case C_\mathfrak p is faithfully flat over B_\mathfrak p (because it is finite and torsion free). Hence B^\wedge _\mathfrak p \to C^\wedge _\mathfrak p is faithfully flat and the same holds after localizing at \mathfrak q. As C^\wedge and hence any localization is (S_2) we conclude that B^\wedge _\mathfrak p is (S_2) by Algebra, Lemma 10.164.5. All in all we find that (S_2) holds for B^\wedge . To prove that B^\wedge is (R_1) we only have to consider primes \mathfrak q \subset B^\wedge with \dim (B^\wedge _\mathfrak q) \leq 1. Since \dim (B^\wedge _\mathfrak q) = \dim (B_\mathfrak p) + \dim (B^\wedge _\mathfrak q/\mathfrak p B^\wedge _\mathfrak q) by Algebra, Lemma 10.112.6 we find that \dim (B_\mathfrak p) \leq 1 and we see that B^\wedge _\mathfrak q \to C^\wedge _\mathfrak q is faithfully flat as before. We conclude using Algebra, Lemma 10.164.6. \square


Comments (0)


Your email address will not be published. Required fields are marked.

In your comment you can use Markdown and LaTeX style mathematics (enclose it like $\pi$). A preview option is available if you wish to see how it works out (just click on the eye in the toolbar).

Unfortunately JavaScript is disabled in your browser, so the comment preview function will not work.

All contributions are licensed under the GNU Free Documentation License.