The Stacks project

Lemma 29.20.5. Let $X$ be a scheme. The following are equivalent:

  1. The scheme $X$ is (quasi-)excellent.

  2. For every affine open $U \subset X$ the ring $\mathcal{O}_ X(U)$ is (quasi-)excellent.

  3. There exists an affine open covering $X = \bigcup U_ i$ such that each $\mathcal{O}_ X(U_ i)$ is (quasi-)excellent.

  4. 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. $\square$

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