Lemma 28.4.3. Let X be a scheme. Let P be a local property of rings. The following are equivalent:
The scheme X is locally P.
For every affine open U \subset X the property P(\mathcal{O}_ X(U)) holds.
There exists an affine open covering X = \bigcup U_ i such that each \mathcal{O}_ X(U_ i) satisfies P.
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 26.11.6 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 28.4.1 (1) we get P(R_{f_ j}). Whereupon P(R) by Definition 28.4.1 (2). \square
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