The Stacks project

Remark 94.15.5. Let $S$ be a scheme. Let $G$ be an abstract group. Let $X$ be an algebraic space over $S$. Let $G \to \text{Aut}_ S(X)$ be a group homomorphism. In this setting we can define $[[X/G]]$ similarly to the above as follows:

  1. An object of $[[X/G]]$ consists of a triple $(U, P, \varphi : P \to X)$ where

    1. $U$ is an object of $(\mathit{Sch}/S)_{fppf}$,

    2. $P$ is a sheaf on $(\mathit{Sch}/U)_{fppf}$ which comes with an action of $G$ that turns it into a torsor under the constant sheaf with value $G$, and

    3. $\varphi : P \to X$ is a $G$-equivariant map of sheaves.

  2. A morphism $(f, g) : (U, P, \varphi ) \to (U', P', \varphi ')$ is given by a morphism of schemes $f : T \to T'$ and a $G$-equivariant isomorphism $g : P \to f^{-1}P'$ such that $\varphi = \varphi ' \circ g$.

In exactly the same manner as above we obtain a functor

\[ [[X/G]] \longrightarrow (\mathit{Sch}/S)_{fppf} \]

which turns $[[X/G]]$ into a stack in groupoids over $(\mathit{Sch}/S)_{fppf}$. The constant sheaf $\underline{G}$ is (provided the cardinality of $G$ is not too large) representable by $G_ S$ on $(\mathit{Sch}/S)_{fppf}$ and this version of $[[X/G]]$ is equivalent to the stack $[[X/G_ S]]$ introduced above.


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