The Stacks project

Definition 26.10.2. Let $X$ be a scheme.

1. A morphism of schemes is called an open immersion if it is an open immersion of locally ringed spaces (see Definition 26.3.1).

2. 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.

3. A morphism of schemes is called a closed immersion if it is a closed immersion of locally ringed spaces (see Definition 26.4.1).

4. 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.

5. 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.