Applications of Nagata's Lemma

Question

In the spirit of this MO question I would like to ask for applications of a (somewhat lesser known?) lemma.

Lemma. (Nagata) Let $R$ be an atomic domain. TFAE:

$R$ is a UFD
There exists a multiplicative submonoid of $R$ generated by prime elements such that $S^{-1}R$ is a UFD

Example application (found in Bill Dubuque's answer to this question):

Claim. $R$ UFD $\implies R[x]$ UFD

Proof. Localizations of UFD's (which don't invert zero) are UFD's. For the hard direction, look at the multiplicative submonoid of $R$ given by $R\setminus \left\{ 0 \right\}$. Since $R$ is a UFD, this monoid is generated by primes. $S^{-1}R=\Bbbk $ is the fraction field of $R$. Now $S^{-1}(R[x])\cong \Bbbk[x]$, but the latter is a PID so in particular a UFD, which implies that $R[x]$ is a UFD itself.

What are some more nice applications of Nagata's lemma?

This is Nagata's criterion for factoriality, it is well known and useful for showing that some rings are UFD's. (Btw, searching for "Nagata UFD" you can find the answers I've posted on M.SE which contains applications of this criterion.) — user26857, Dec 10 '15 at 08:49
@user26857 thanks for the information. If you'd like to post some of your favorite applications that would be cool. — Arrow, Dec 10 '15 at 12:33

score 1 · Answer 1 · edited Dec 10 '15 at 08:50

There is a classic algebraic geometry exercise that uses Nagata's lemma (cf. Vakil's algebraic geometry notes, Exercise 14.2.V).

Problem. If $n\ge 5$, then the class group of the quadric cone

$$Q_n=V(x_1^2+\cdots +x_n^2)\subset \mathbb A^n$$

is trivial.

The solution Vakil suggests is to apply Proposition 6.2 in Hartshorne and show that the coordinate ring is a UFD using Nagata's lemma. Another proof is sketched in Exercises II.6.4 and II.6.5 in Hartshorne.

Applications of Nagata's Lemma

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