The Stacks project

Proof. First, assume that $g : U' \to U$ satisfies (1) and (2). Property (1) implies that $[U'/R'] \to [U/R]$ is an equivalence, see Groupoids in Spaces, Lemma 78.25.2. By Theorem 97.17.2 the quotient stack $[U'/R']$ is an algebraic stack. Hence $[U/R]$ is an algebraic stack too, see Algebraic Stacks, Lemma 94.12.4.

Conversely, assume that $[U/R]$ is an algebraic stack. We may choose a scheme $W$ and a surjective smooth $1$-morphism

By the $2$-Yoneda lemma (Algebraic Stacks, Section 94.5) this corresponds to an object $\xi$ of $[U/R]$ over $W$. By the description of $[U/R]$ in Groupoids in Spaces, Lemma 78.24.1 we can find a surjective, flat, locally finitely presented morphism $b : U' \to W$ of schemes such that $\xi ' = b^*\xi$ corresponds to a morphism $g : U' \to U$. Note that the $1$-morphism

corresponding to $\xi '$ is surjective, flat, and locally of finite presentation, see Algebraic Stacks, Lemma 94.10.5. Hence $(\mathit{Sch}/U')_{fppf} \times _{[U/R]} (\mathit{Sch}/U')_{fppf}$ which is represented by the algebraic space

(see Groupoids in Spaces, Lemma 78.22.1 for the first equality; the second is the definition of restriction) is flat and locally of finite presentation over $U'$ via both $s'$ and $t'$ (by base change, see Algebraic Stacks, Lemma 94.10.6). By this description of $R'$ and by Algebraic Stacks, Lemma 94.16.1 we obtain a canonical fully faithful $1$-morphism $[U'/R'] \to [U/R]$. This $1$-morphism is essentially surjective because $f'$ is flat, locally of finite presentation, and surjective (see Stacks, Lemma 8.4.8); another way to prove this is to use Algebraic Stacks, Remark 94.16.3. Finally, we can use Groupoids in Spaces, Lemma 78.25.2 to conclude that the composition $U' \times _{g, U, t} R \to R \xrightarrow {s} U$ is a surjection of sheaves. $\square$

Proof. This lemma is a special case of Lemma 97.18.1. Namely, the quotient stack $[X/G]$ is by Groupoids in Spaces, Definition 78.20.1 equal to the quotient stack $[X/G \times _ B X]$ of the groupoid in algebraic spaces $(X, G \times _ B X, s, t, c)$ associated to the group action in Groupoids in Spaces, Lemma 78.15.1. There is one small observation that is needed to get condition (1). Namely, the morphism $s : G \times _ B X \to X$ is the second projection and the morphism $t : G \times _ B X \to X$ is the action morphism $a$. Hence the morphism $h : U' \times _{g, U, t} R \to R \xrightarrow {s} U$ from Lemma 97.18.1 corresponds to the morphism

in the current setting. However, because of the symmetry given by the inverse of $G$ this morphism is isomorphic to the morphism

of the statement of the lemma. Details omitted. $\square$

Proof. If $G$ is flat and locally of finite presentation over $B$, then $[B/G]$ is an algebraic stack by Theorem 97.17.2.

Conversely, assume that $[B/G]$ is an algebraic stack. By Lemma 97.18.2 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

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 74.8.1 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 74.11.13 and 74.11.10 we conclude that $G \to B$ is flat and locally of finite presentation. $\square$