Schemes can be glued to give new schemes.

Lemma 26.14.2. In Lemma 26.14.1 above, assume that all $X_ i$ are schemes. Then the resulting locally ringed space $X$ is a scheme.

Proof. This is clear since each of the $U_ i$ is a scheme and hence every $x \in X$ has an affine neighbourhood. $\square$

Comment #840 by on

Suggested slogan: Schemes are closed under gluing in the category of locally ringed spaces.

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