The Stacks project

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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