The Stacks project

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

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

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