The Stacks project

77.24 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 77.20.1 (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

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

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

such that

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

    \[ s \circ r_{ij} = u_ i \circ \text{pr}_0 \quad \text{and}\quad t \circ r_{ij} = u_ j \circ \text{pr}_1, \]
  2. 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

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

    \[ u_ i = s \circ r_ i \quad \text{and}\quad u'_ i = t \circ r_ i \]
  2. $(\beta )$ 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 77.24.1. Assume $B \to S$ and $(U, R, s, t, c)$ are as in Definition 77.20.1 (1). Let $\pi : \mathcal{S}_ U \to [U/R]$ be as in Lemma 77.20.2. Let $T$ be a scheme over $S$.

  1. 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 $u_ i \in U(T_ i)$,

  2. 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 $r_{ij} : T_ i \times _ T T_ j \to R$,

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

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

  5. 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

    \[ \mathop{\mathrm{Mor}}\nolimits _{[U/R]_ T}(x, y) \longleftrightarrow \left\{ \begin{matrix} \text{morphisms }(u_ i, r_{ij}) \to (u'_ i, r'_{ij}) \\ \text{of }[U/R]\text{-descent data} \end{matrix} \right\} \]
  6. 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 77.22.1. 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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