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

Proof. We may assume that is affine. If
is finite then is is an isomorphism over a
dense open subset.

Conversely, assume that is normal. We apply
(Claim 2) to get a finite partial normalization
that is an isomorphism over
such that is either regular (of dim 1) or has depth at all points of . Thus is normal. \qed

{ Claim 2.}
Let be a reduced, Noetherian scheme that is locally N-1. Let be a finite set of points and
a non-zerodivisor. Then there is a
finite partial normalization
such that is either regular (of dim 1) or has depth at all preimages of the and
at all points of .
We can also assume that is an isomorphism over
.

\medskip

Proof. Pick any , take the
punctual normalization
(=maximal partial normalization that is isomorphism outside )
and then extend it to a partial normalization that is an
isomorphism over .
Using this procedure inductively, we take care of the first part.
(Note: the N-1 condition is inherited by .)

To ensure the second part, we first apply this argument to the generic points of . Thus we may assume that is regular at all generic points of . The points of where has depth correspond to the embedded points of . 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 10.155.15 which has essentially the same proof as yours, but was in Section 10.156 (Nagata rings). I have moved that lemma and Lemma 10.155.14 which it rests on to this Section 10.155 (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!

Your email address will not be published.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.

## Comments (2)

Comment #1539 by kollar on

Comment #1542 by Johan on

There are also: