Lemma 26.4.2. Let $f : Z \to X$ be a morphism of locally ringed spaces. In order for $f$ to be a closed immersion it suffices that there exists an open covering $X = \bigcup U_ i$ such that each $f : f^{-1}U_ i \to U_ i$ is a closed immersion.

Proof. Omitted. $\square$

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