The Stacks project

77.22 The 2-cartesian square of a quotient stack

In this section we compute the $\mathit{Isom}$-sheaves for a quotient stack and we deduce that the defining diagram of a quotient stack is a $2$-fibre product.

Lemma 77.22.1. Assume $B \to S$, $(U, R, s, t, c)$ and $\pi : \mathcal{S}_ U \to [U/R]$ are as in Lemma 77.20.2. Let $S'$ be a scheme over $S$. Let $x, y \in \mathop{\mathrm{Ob}}\nolimits ([U/R]_{S'})$ be objects of the quotient stack over $S'$. If $x = \pi (x')$ and $y = \pi (y')$ for some morphisms $x', y' : S' \to U$, then

\[ \mathit{Isom}(x, y) = S' \times _{(y', x'), U \times _ S U} R \]

as sheaves over $S'$.

Proof. Let $[U/_{\! p}R]$ be the category fibred in groupoids associated to the presheaf in groupoids (77.20.0.1) as in the proof of Lemma 77.20.2. By construction the sheaf $\mathit{Isom}(x, y)$ is the sheaf associated to the presheaf $\mathit{Isom}(x', y')$. On the other hand, by definition of morphisms in $[U/_{\! p}R]$ we have

\[ \mathit{Isom}(x', y') = S' \times _{(y', x'), U \times _ S U} R \]

and the right hand side is an algebraic space, therefore a sheaf. $\square$

Lemma 77.22.2. Assume $B \to S$, $(U, R, s, t, c)$, and $\pi : \mathcal{S}_ U \to [U/R]$ are as in Lemma 77.20.2. The $2$-commutative square

\[ \xymatrix{ \mathcal{S}_ R \ar[r]_ s \ar[d]_ t & \mathcal{S}_ U \ar[d]^\pi \\ \mathcal{S}_ U \ar[r]^-\pi & [U/R] } \]

of Lemma 77.20.3 is a $2$-fibre product of stacks in groupoids of $(\mathit{Sch}/S)_{fppf}$.

Proof. According to Stacks, Lemma 8.5.6 the lemma makes sense. It also tells us that we have to show that the functor

\[ \mathcal{S}_ R \longrightarrow \mathcal{S}_ U \times _{[U/R]} \mathcal{S}_ U \]

which maps $r : T \to R$ to $(T, t(r), s(r), \alpha (r))$ is an equivalence, where the right hand side is the $2$-fibre product as described in Categories, Lemma 4.32.3. This is, after spelling out the definitions, exactly the content of Lemma 77.22.1. (Alternative proof: Work out the meaning of Lemma 77.21.2 in this situation will give you the result also.) $\square$

Lemma 77.22.3. Assume $B \to S$ and $(U, R, s, t, c)$ are as in Definition 77.20.1 (1). For any scheme $T$ over $S$ and objects $x, y$ of $[U/R]$ over $T$ the sheaf $\mathit{Isom}(x, y)$ on $(\mathit{Sch}/T)_{fppf}$ has the following property: There exists a fppf covering $\{ T_ i \to T\} _{i \in I}$ such that $\mathit{Isom}(x, y)|_{(\mathit{Sch}/T_ i)_{fppf}}$ is representable by an algebraic space.

Proof. Follows immediately from Lemma 77.22.1 and the fact that both $x$ and $y$ locally in the fppf topology come from objects of $\mathcal{S}_ U$ by construction of the quotient stack. $\square$


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