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Tag 01RD

Chapter 28: Morphisms of Schemes > Section 28.7: Scheme theoretic closure and density

Lemma 28.7.3. Let $X$ be a scheme. Let $U \subset X$ be an open subscheme. If the inclusion morphism $U \to X$ is quasi-compact, then $U$ is scheme theoretically dense in $X$ if and only if the scheme theoretic closure of $U$ in $X$ is $X$.

Proof. Follows from Lemma 28.6.3 part (3). $\square$

    The code snippet corresponding to this tag is a part of the file morphisms.tex and is located in lines 1040–1047 (see updates for more information).

    \begin{lemma}
    \label{lemma-scheme-theoretically-dense-quasi-compact}
    Let $X$ be a scheme.
    Let $U \subset X$ be an open subscheme.
    If the inclusion morphism $U \to X$ is quasi-compact, then $U$
    is scheme theoretically dense in $X$ if and only if the scheme theoretic
    closure of $U$ in $X$ is $X$.
    \end{lemma}
    
    \begin{proof}
    Follows from Lemma \ref{lemma-quasi-compact-scheme-theoretic-image} part (3).
    \end{proof}

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