Definition 26.10.2. Let $X$ be a scheme.

A morphism of schemes is called an

*open immersion*if it is an open immersion of locally ringed spaces (see Definition 26.3.1).An

*open subscheme*of $X$ is an open subspace of $X$ in the sense of Definition 26.3.3; an open subscheme of $X$ is a scheme by Lemma 26.9.2.A morphism of schemes is called a

*closed immersion*if it is a closed immersion of locally ringed spaces (see Definition 26.4.1).A

*closed subscheme*of $X$ is a closed subspace of $X$ in the sense of Definition 26.4.4; a closed subscheme is a scheme by Lemma 26.10.1.A morphism of schemes $f : X \to Y$ is called an

*immersion*, or a*locally closed immersion*if it can be factored as $j \circ i$ where $i$ is a closed immersion and $j$ is an open immersion.

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