The Stacks project

The following is a converse to 032G

Claim 1. Let $X$ be a Noetherian, integral scheme whose local rings are N-1. Then $X$ is N-1 iff $X$ contains a dense, open, normal subset.

Proof. We may assume that $X$ is affine. If $X^{n}\to X$ is finite then is is an isomorphism over a dense open subset.

Conversely, assume that $X\setminus(g=0)$ is normal. We apply (Claim 2) to get a finite partial normalization $p:Y\to X$ that is an isomorphism over $X\setminus (g=0)$ such that $Y$ is either regular (of dim 1) or has depth $\geq 2$ at all points of $(p^*g=0)$. Thus $Y$ is normal. \qed

{ Claim 2.} Let $X$ be a reduced, Noetherian scheme that is locally N-1. Let $x_i\in X$ be a finite set of points and $g\in \o_X$ a non-zerodivisor. Then there is a finite partial normalization $p:Y\to X$ such that $Y$ is either regular (of dim 1) or has depth $\geq 2$ at all preimages of the $x_i$ and at all points of $(p^*g=0)$. We can also assume that $p$ is an isomorphism over $X\setminus (g=0)\cup\bigcup_i\bar x_i$.

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Proof. Pick any $x\in X$, take the punctual normalization $\bigl(x^{\rm pn}, X^{\rm pn}\bigr)\to (x, X)$ (=maximal partial normalization that is isomorphism outside $x$) and then extend it to a partial normalization $X'\to X$ that is an isomorphism over $X\setminus\bar x$. Using this procedure inductively, we take care of the first part. (Note: the N-1 condition is inherited by $X'$.)

To ensure the second part, we first apply this argument to the generic points of $(g=0)$. Thus we may assume that $X$ is regular at all generic points of $(g=0)$. The points of $(g=0)$ where $X$ has depth $\leq 1$ correspond to the embedded points of $(g=0)$. We first remove the generic points of the set of embedded points and then repeat the argument. \qed

Dear kollar, the nontrivial direction of Claim 1 is Lemma 10.161.15 which has essentially the same proof as yours, but was in Section 10.162 (Nagata rings). I have moved that lemma and Lemma 10.161.14 which it rests on to this Section 10.161 (Japanese rings). Sorry for the misplaced lemmas!

Claim 2, which can be used to improve locally N-1 but non-N-1 rings is missing for the moment. We will add this in this section if we ever need it. Or if you'd like us to add it so you can refer to it, then let us know.

You can find the edits to the tex file here. Thanks!