The Stacks project

Lemma 28.14.3. Let $X$ be a scheme. The following are equivalent:

1. The scheme $X$ is a G-scheme.

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

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

4. There exists an open covering $X = \bigcup X_ j$ such that each open subscheme $X_ j$ is a G-scheme.

Moreover, if $X$ is a G-scheme then every open subscheme is a G-scheme.