The Stacks Project

Tag: 01RB

This tag has label morphisms-definition-scheme-theoretically-dense and it points to

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Definition 25.7.1. Let $X$ be a scheme. Let $U \subset X$ be an open subscheme.
1. The scheme theoretic image of the morphism $U \to X$ is called the scheme theoretic closure of $U$ in $X$.
2. We say $U$ is scheme theoretically dense in $X$ if for every open $V \subset X$ the scheme theoretic closure of $U \cap V$ in $V$ is equal to $V$.

\begin{definition}
\label{definition-scheme-theoretically-dense}
Let $X$ be a scheme. Let $U \subset X$ be an open subscheme.
\begin{enumerate}
\item The scheme theoretic image of the morphism $U \to X$
is called the {\it scheme theoretic closure of $U$ in $X$}.
\item We say $U$ is {\it scheme theoretically dense in $X$}
if for every open $V \subset X$ the scheme theoretic closure
of $U \cap V$ in $V$ is equal to $V$.
\end{enumerate}
\end{definition}


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