# The Stacks Project

## Tag: 03XP

This tag has label spaces-morphisms-definition-locally-finite-presentation and it points to

The corresponding content:

Definition 45.26.1. Let $S$ be a scheme. Let $X \to Y$ be a morphism of algebraic spaces over $S$.
1. We say $f$ is locally of finite presentation if the equivalent conditions of Lemma 45.21.1 hold with $\mathcal{P} =$''locally of finite presentation''.
2. Let $x \in |X|$. We say $f$ is of finite presentation at $x$ if there exists an open neighbourhood $X' \subset X$ of $x$ such that $f|_{X'} : X' \to Y$ is locally of finite presentation\footnote{It seems awkward to use ''locally of finite presentation at $x$'', but the current terminology may be misleading in the sense that ''of finite presentation at $x$'' does not mean that there is an open neighbourhood $X' \subset X$ such that $f|_{X'}$ is of finite presentation.}.
3. A morphism of algebraic spaces $f : X \to Y$ is of finite presentation if it is locally of finite presentation, quasi-compact and quasi-separated.

\begin{definition}
\label{definition-locally-finite-presentation}
Let $S$ be a scheme.
Let $X \to Y$ be a morphism of algebraic spaces over $S$.
\begin{enumerate}
\item We say $f$ is {\it locally of finite presentation} if
the equivalent conditions of
Lemma \ref{lemma-local-source-target}
hold with $\mathcal{P} =$locally of finite presentation''.
\item Let $x \in |X|$. We say $f$ is of {\it finite presentation at $x$}
if there exists an open neighbourhood $X' \subset X$ of $x$ such
that $f|_{X'} : X' \to Y$ is locally of finite
presentation\footnote{It seems awkward to use locally of finite presentation
at $x$'', but the current terminology may be misleading in the sense that
of finite presentation at $x$'' does {\bf not} mean that there is
an open neighbourhood $X' \subset X$ such that $f|_{X'}$ is of finite
presentation.}.
\item A morphism of algebraic spaces $f : X \to Y$ is
{\it of finite presentation}
if it is locally of finite presentation, quasi-compact and
quasi-separated.
\end{enumerate}
\end{definition}


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