The Stacks project

Lemma 27.4.3. Let $X$ be a scheme. Let $P$ be a local property of rings. The following are equivalent:

  1. The scheme $X$ is locally $P$.

  2. For every affine open $U \subset X$ the property $P(\mathcal{O}_ X(U))$ holds.

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

  4. There exists an open covering $X = \bigcup X_ j$ such that each open subscheme $X_ j$ is locally $P$.

Moreover, if $X$ is locally $P$ then every open subscheme is locally $P$.

Proof. Of course (1) $\Leftrightarrow $ (3) and (2) $\Rightarrow $ (1). If (3) $\Rightarrow $ (2), then the final statement of the lemma holds and it follows easily that (4) is also equivalent to (1). Thus we show (3) $\Rightarrow $ (2).

Let $X = \bigcup U_ i$ be an affine open covering, say $U_ i = \mathop{\mathrm{Spec}}(R_ i)$. Assume $P(R_ i)$. Let $\mathop{\mathrm{Spec}}(R) = U \subset X$ be an arbitrary affine open. By Schemes, Lemma 25.11.6 there exists a standard covering of $U = \mathop{\mathrm{Spec}}(R)$ by standard opens $D(f_ j)$ such that each ring $R_{f_ j}$ is a principal localization of one of the rings $R_ i$. By Definition 27.4.1 (1) we get $P(R_{f_ j})$. Whereupon $P(R)$ by Definition 27.4.1 (2). $\square$


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