Lemma 89.2.2. Let $(A, \mathfrak m, \kappa )$ be a Noetherian local ring. Let $g : X \to Y$ be a morphism in the category (89.2.0.1). If the induced morphism $X_\kappa \to Y_\kappa $ of special fibres is a closed immersion, then $g$ is a closed immersion.
Proof. This is a special case of More on Morphisms of Spaces, Lemma 76.49.3. $\square$
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