## Tag `06PL`

Chapter 87: Criteria for Representability > Section 87.18: When is a quotient stack algebraic?

Lemma 87.18.3. Let $S$ be a scheme and let $B$ be an algebraic space over $S$. Let $G$ be a group algebraic space over $B$. Endow $B$ with the trivial action of $G$. Then the quotient stack $[B/G]$ is an algebraic stack if and only if $G$ is flat and locally of finite presentation over $B$.

Proof.If $G$ is flat and locally of finite presentation over $B$, then $[B/G]$ is an algebraic stack by Theorem 87.17.2.Conversely, assume that $[B/G]$ is an algebraic stack. By Lemma 87.18.2 and because the action is trivial, we see there exists an algebraic space $B'$ and a morphism $B' \to B$ such that (1) $B' \to B$ is a surjection of sheaves and (2) the projections $$ B' \times_B G \times_B B' \to B' $$ are flat and locally of finite presentation. Note that the base change $B' \times_B G \times_B B' \to G \times_B B'$ of $B' \to B$ is a surjection of sheaves also. Thus it follows from Descent on Spaces, Lemma 65.7.1 that the projection $G \times_B B' \to B'$ is flat and locally of finite presentation. By (1) we can find an fppf covering $\{B_i \to B\}$ such that $B_i \to B$ factors through $B' \to B$. Hence $G \times_B B_i \to B_i$ is flat and locally of finite presentation by base change. By Descent on Spaces, Lemmas 65.10.13 and 65.10.10 we conclude that $G \to B$ is flat and locally of finite presentation. $\square$

The code snippet corresponding to this tag is a part of the file `criteria.tex` and is located in lines 3211–3218 (see updates for more information).

```
\begin{lemma}
\label{lemma-BG-algebraic}
Let $S$ be a scheme and let $B$ be an algebraic space over $S$.
Let $G$ be a group algebraic space over $B$.
Endow $B$ with the trivial action of $G$.
Then the quotient stack $[B/G]$ is an algebraic stack
if and only if $G$ is flat and locally of finite presentation over $B$.
\end{lemma}
\begin{proof}
If $G$ is flat and locally of finite presentation over $B$, then
$[B/G]$ is an algebraic stack by
Theorem \ref{theorem-flat-groupoid-gives-algebraic-stack}.
\medskip\noindent
Conversely, assume that $[B/G]$ is an algebraic stack. By
Lemma \ref{lemma-group-quotient-algebraic}
and because the action is trivial, we see
there exists an algebraic space $B'$ and a morphism
$B' \to B$ such that (1) $B' \to B$ is a surjection
of sheaves and (2) the projections
$$
B' \times_B G \times_B B' \to B'
$$
are flat and locally of finite presentation. Note that the base change
$B' \times_B G \times_B B' \to G \times_B B'$ of $B' \to B$
is a surjection of sheaves also. Thus it follows from
Descent on Spaces, Lemma \ref{spaces-descent-lemma-curiosity}
that the projection $G \times_B B' \to B'$ is flat and locally
of finite presentation. By (1) we can find an fppf covering
$\{B_i \to B\}$ such that $B_i \to B$ factors through $B' \to B$.
Hence $G \times_B B_i \to B_i$ is flat and locally of finite presentation
by base change. By
Descent on Spaces, Lemmas
\ref{spaces-descent-lemma-descending-property-flat} and
\ref{spaces-descent-lemma-descending-property-locally-finite-presentation}
we conclude that $G \to B$ is flat and locally of finite presentation.
\end{proof}
```

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