Remark 110.33.6. When $(X, {\mathcal O}_ X)$ is a ringed space and $U \subset X$ is an open subset then $(U, {\mathcal O}_ X|_ U)$ is a ringed space. Notation: ${\mathcal O}_ U = {\mathcal O}_ X|_ U$. There is a canonical morphisms of ringed spaces

$j : (U, {\mathcal O}_ U) \longrightarrow (X, {\mathcal O}_ X).$

If $(X, {\mathcal O}_ X)$ is a locally ringed space, so is $(U, {\mathcal O}_ U)$ and $j$ is a morphism of locally ringed spaces. If $(X, {\mathcal O}_ X)$ is a scheme so is $(U, {\mathcal O}_ U)$ and $j$ is a morphism of schemes. We say that $(U, {\mathcal O}_ U)$ is an open subscheme of $(X, {\mathcal O}_ X)$ and that $j$ is an open immersion. More generally, any morphism $j' : (V, {\mathcal O}_ V) \to (X, {\mathcal O}_ X)$ that is isomorphic to a morphism $j : (U, {\mathcal O}_ U) \to (X, {\mathcal O}_ X)$ as above is called an open immersion.

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