## Tag `03I8`

Chapter 59: Decent Algebraic Spaces > Section 59.6: Reasonable and decent algebraic spaces

Definition 59.6.1. Let $S$ be a scheme. Let $X$ be an algebraic space over $S$.

- We say $X$ is
decentif for every point $x \in X$ the equivalent conditions of Lemma 59.4.5 hold, in other words property $(\gamma)$ of Lemma 59.5.1 holds.- We say $X$ is
reasonableif the equivalent conditions of Lemma 59.4.6 hold, in other words property $(\delta)$ of Lemma 59.5.1 holds.- We say $X$ is
very reasonableif the equivalent conditions of Lemma 59.4.7 hold, i.e., property $(\epsilon)$ of Lemma 59.5.1 holds.

The code snippet corresponding to this tag is a part of the file `decent-spaces.tex` and is located in lines 1030–1052 (see updates for more information).

```
\begin{definition}
\label{definition-very-reasonable}
Let $S$ be a scheme.
Let $X$ be an algebraic space over $S$.
\begin{enumerate}
\item We say $X$ is {\it decent} if for every point $x \in X$ the equivalent
conditions of
Lemma \ref{lemma-UR-finite-above-x}
hold, in other words property $(\gamma)$ of
Lemma \ref{lemma-bounded-fibres}
holds.
\item We say $X$ is {\it reasonable} if the equivalent conditions of
Lemma \ref{lemma-U-universally-bounded}
hold, in other words property $(\delta)$ of
Lemma \ref{lemma-bounded-fibres}
holds.
\item We say $X$ is {\it very reasonable} if the equivalent conditions of
Lemma \ref{lemma-characterize-very-reasonable}
hold, i.e., property $(\epsilon)$ of
Lemma \ref{lemma-bounded-fibres}
holds.
\end{enumerate}
\end{definition}
```

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