## 78.24 Explicit description of quotient stacks

In order to formulate the result we need to introduce some notation. Assume $B \to S$ and $(U, R, s, t, c)$ are as in Definition 78.20.1 (1). Let $T$ be a scheme over $S$. Let $\mathcal{T} = \{ T_ i \to T\} _{i \in I}$ be an fppf covering. A *$[U/R]$-descent datum* relative to $\mathcal{T}$ is given by a system $(u_ i, r_{ij})$ where

for each $i$ a morphism $u_ i : T_ i \to U$, and

for each $i, j$ a morphism $r_{ij} : T_ i \times _ T T_ j \to R$

such that

as morphisms $T_ i \times _ T T_ j \to U$ we have

\[ s \circ r_{ij} = u_ i \circ \text{pr}_0 \quad \text{and}\quad t \circ r_{ij} = u_ j \circ \text{pr}_1, \]

as morphisms $T_ i \times _ T T_ j \times _ T T_ k \to R$ we have

\[ c \circ (r_{jk} \circ \text{pr}_{12}, r_{ij} \circ \text{pr}_{01}) = r_{ik} \circ \text{pr}_{02}. \]

A *morphism $(u_ i, r_{ij}) \to (u'_ i, r'_{ij})$ between two $[U/R]$-descent data* over the same covering $\mathcal{T}$ is a collection $(r_ i : T_ i \to R)$ such that

$(\alpha )$ as morphisms $T_ i \to U$ we have

\[ u_ i = s \circ r_ i \quad \text{and}\quad u'_ i = t \circ r_ i \]

$(\beta )$ as morphisms $T_ i \times _ T T_ j \to R$ we have

\[ c \circ (r'_{ij}, r_ i \circ \text{pr}_0) = c \circ (r_ j \circ \text{pr}_1, r_{ij}). \]

There is a natural composition law on morphisms of descent data relative to a fixed covering and we obtain a category of descent data. This category is a groupoid. Finally, if $\mathcal{T}' = \{ T'_ j \to T\} _{j \in J}$ is a second fppf covering which refines $\mathcal{T}$ then there is a notion of pullback of descent data. This is particularly easy to describe explicitly in this case. Namely, if $\alpha : J \to I$ and $\varphi _ j : T'_ j \to T_{\alpha (i)}$ is the morphism of coverings, then the pullback of the descent datum $(u_ i, r_{ii'})$ is simply

\[ (u_{\alpha (i)} \circ \varphi _ j, r_{\alpha (j)\alpha (j')} \circ \varphi _ j \times \varphi _{j'}). \]

Pullback defined in this manner defines a functor from the category of descent data over $\mathcal{T}$ to the category of descend data over $\mathcal{T}'$.

Lemma 78.24.1. Assume $B \to S$ and $(U, R, s, t, c)$ are as in Definition 78.20.1 (1). Let $\pi : \mathcal{S}_ U \to [U/R]$ be as in Lemma 78.20.2. Let $T$ be a scheme over $S$.

for every object $x$ of the fibre category $[U/R]_ T$ there exists an fppf covering $\{ f_ i : T_ i \to T\} _{i \in I}$ such that $f_ i^*x \cong \pi (u_ i)$ for some $u_ i \in U(T_ i)$,

the composition of the isomorphisms

\[ \pi (u_ i \circ \text{pr}_0) = \text{pr}_0^*\pi (u_ i) \cong \text{pr}_0^*f_ i^*x \cong \text{pr}_1^*f_ j^*x \cong \text{pr}_1^*\pi (u_ j) = \pi (u_ j \circ \text{pr}_1) \]

are of the form $\pi (r_{ij})$ for certain morphisms $r_{ij} : T_ i \times _ T T_ j \to R$,

the system $(u_ i, r_{ij})$ forms a $[U/R]$-descent datum as defined above,

any $[U/R]$-descent datum $(u_ i, r_{ij})$ arises in this manner,

if $x$ corresponds to $(u_ i, r_{ij})$ as above, and $y \in \mathop{\mathrm{Ob}}\nolimits ([U/R]_ T)$ corresponds to $(u'_ i, r'_{ij})$ then there is a canonical bijection

\[ \mathop{\mathrm{Mor}}\nolimits _{[U/R]_ T}(x, y) \longleftrightarrow \left\{ \begin{matrix} \text{morphisms }(u_ i, r_{ij}) \to (u'_ i, r'_{ij})
\\ \text{of }[U/R]\text{-descent data}
\end{matrix} \right\} \]

this correspondence is compatible with refinements of fppf coverings.

**Proof.**
Statement (1) is part of the construction of the stackyfication. Part (2) follows from Lemma 78.22.1. We omit the verification of (3). Part (4) is a translation of the fact that in a stack all descent data are effective. We omit the verifications of (5) and (6).
$\square$

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