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Tag 01HL

Chapter 25: Schemes > Section 25.4: Closed immersions of locally ringed spaces

Lemma 25.4.2. 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. $\square$

    The code snippet corresponding to this tag is a part of the file schemes.tex and is located in lines 355–361 (see updates for more information).

    \begin{lemma}
    \label{lemma-closed-local-target}
    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.
    \end{lemma}
    
    \begin{proof}
    Omitted.
    \end{proof}

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