We recall that a ring is quasi-excellent if it is a G-ring and J-2. A ring is excellent if it is quasi-excellent and universally catenary.
Definition 29.20.1. Let $X$ be a scheme.
We say $X$ is quasi-excellent if for every $x \in X$ there exists an affine open neighbourhood $x \in U \subset X$ such that the ring $\mathcal{O}_ X(U)$ is quasi-excellent (see More on Algebra, Definition 15.53.1).
We say $X$ is excellent if for every $x \in X$ there exists an affine open neighbourhood $x \in U \subset X$ such that the ring $\mathcal{O}_ X(U)$ is excellent (see More on Algebra, Definition 15.53.1).
Quasi-excellent schemes are locally Noetherian (G-rings are Noetherian).
Lemma 29.20.2. Let $X$ be a scheme. The following are equivalent
$X$ is quasi-excellent, and
$X$ is a G-scheme and J-2.
Proof. The implication (1) $\Rightarrow $ (2) follows on combining the definition of quasi-excellent schemes (Definition 29.20.1), the definition of quasi-excellent rings (More on Algebra, Definition 15.53.1), the definition of G-schemes (Properties, Definition 28.14.1), and Lemma 29.19.2. Conversely, if $X$ is a G-scheme and J-2, then for any $x \in X$ we can find an affine open neighbourhood $x \in U \subset X$ such that $\mathcal{O}_ X(U)$ is a G-ring. By Lemma 29.19.2 the ring $\mathcal{O}_ X(U)$ is also J-2, whence quasi-excellent. $\square$
Lemma 29.20.3. A quasi-excellent scheme is Nagata.
Proof. See More on Algebra, Lemma 15.53.5. $\square$
Lemma 29.20.4. Let $X$ be a scheme. The following are equivalent
$X$ is excellent, and
$X$ is quasi-excellent and universally catenary.
Proof. Assume (1). Since excellent rings are quasi-excellent, it is clear that $X$ is quasi-excellent. Since excellent rings are universally catenary, it also follows that $X$ is universally catenary by Lemma 29.17.2. Assume (2). Let $x \in X$. Since $X$ is quasi-excellent there exists an affine open neighbourhood $x \in U \subset X$ such that $\mathcal{O}_ X(U)$ is quasi-excellent. Since $X$ is universally category, the ring $\mathcal{O}_ X(U)$ is also universally catenary by Lemma 29.17.2, whence excellent. $\square$
Lemma 29.20.5. Let $X$ be a scheme. The following are equivalent:
The scheme $X$ is (quasi-)excellent.
For every affine open $U \subset X$ the ring $\mathcal{O}_ X(U)$ is (quasi-)excellent.
There exists an affine open covering $X = \bigcup U_ i$ such that each $\mathcal{O}_ X(U_ i)$ is (quasi-)excellent.
There exists an open covering $X = \bigcup X_ j$ such that each open subscheme $X_ j$ is (quasi-)excellent.
Moreover, if $X$ is (quasi-)excellent then every open subscheme is (quasi-)excellent.
Proof. By Lemma 29.20.2 being quasi-excellent is the same as being a G-scheme and J-2. Thus for the quasi-excellent case, the lemma follows from Lemma 29.19.2 and Properties, Lemma 28.14.3. For the excellent case, use Lemma 29.20.4, the quasi-excellent case, and Lemma 29.17.2. $\square$
Lemma 29.20.6. Let $f : X \to S$ be a morphism. If $S$ is (quasi-)excellent and $f$ locally of finite type then $X$ is (quasi-)excellent.
Proof. See More on Algebra, Lemma 15.53.2. $\square$
Lemma 29.20.7. The following types of schemes are excellent.
Any scheme locally of finite type over a field.
Any scheme locally of finite type over a Noetherian complete local ring.
Any scheme locally of finite type over $\mathbf{Z}$.
Any scheme locally of finite type over a Dedekind ring of characteristic zero.
Proof. By Lemmas 29.20.5 and 29.20.6 we only need to show that the rings mentioned above are excellent. For this see More on Algebra, Proposition 15.53.3. $\square$
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