Lemma 21.11.1. Let \mathcal{C} be a site. Let p : \mathcal{S} \to \mathcal{C} be a gerbe over a site whose automorphism sheaves are abelian. Let \mathcal{G} be the sheaf of abelian groups constructed in Stacks, Lemma 8.11.8. Let U be an object of \mathcal{C} such that
there exists a cofinal system of coverings \{ U_ i \to U\} of U in \mathcal{C} such that H^1(U_ i, \mathcal{G}) = 0 and H^1(U_ i \times _ U U_ j, \mathcal{G}) = 0 for all i, j, and
H^2(U, \mathcal{G}) = 0.
Then there exists an object of \mathcal{S} lying over U.
Proof.
By Stacks, Definition 8.11.1 there exists a covering \mathcal{U} = \{ U_ i \to U\} and x_ i in \mathcal{S} lying over U_ i. Write U_{ij} = U_ i \times _ U U_ j. By (1) after refining the covering we may assume that H^1(U_ i, \mathcal{G}) = 0 and H^1(U_{ij}, \mathcal{G}) = 0. Consider the sheaf
\mathcal{F}_{ij} = \mathit{Isom}(x_ i|_{U_{ij}}, x_ j|_{U_{ij}})
on \mathcal{C}/U_{ij}. Since \mathcal{G}|_{U_{ij}} = \mathit{Aut}(x_ i|_{U_{ij}}) we see that there is an action
\mathcal{G}|_{U_{ij}} \times \mathcal{F}_{ij} \to \mathcal{F}_{ij}
by precomposition. It is clear that \mathcal{F}_{ij} is a pseudo \mathcal{G}|_{U_{ij}}-torsor and in fact a torsor because any two objects of a gerbe are locally isomorphic. By our choice of the covering and by Lemma 21.4.3 these torsors are trivial (and hence have global sections by Lemma 21.4.2). In other words, we can choose isomorphisms
\varphi _{ij} : x_ i|_{U_{ij}} \longrightarrow x_ j|_{U_{ij}}
To find an object x over U we are going to massage our choice of these \varphi _{ij} to get a descent datum (which is necessarily effective as p : \mathcal{S} \to \mathcal{C} is a stack). Namely, the obstruction to being a descent datum is that the cocycle condition may not hold. Namely, set U_{ijk} = U_ i \times _ U U_ j \times _ U U_ k. Then we can consider
g_{ijk} = \varphi _{ik}^{-1}|_{U_{ijk}} \circ \varphi _{jk}|_{U_{ijk}} \circ \varphi _{ij}|_{U_{ijk}}
which is an automorphism of x_ i over U_{ijk}. Thus we may and do consider g_{ijk} as a section of \mathcal{G} over U_{ijk}. A computation (omitted) shows that (g_{i_0i_1i_2}) is a 2-cocycle in the Čech complex {\check C}^\bullet (\mathcal{U}, \mathcal{G}) of \mathcal{G} with respect to the covering \mathcal{U}. By the spectral sequence of Lemma 21.10.6 and since H^1(U_ i, \mathcal{G}) = 0 for all i we see that {\check H}^2(\mathcal{U}, \mathcal{G}) \to H^2(U, \mathcal{G}) is injective. Hence (g_{i_0i_1i_2}) is a coboundary by our assumption that H^2(U, \mathcal{G}) = 0. Thus we can find sections g_{ij} \in \mathcal{G}(U_{ij}) such that g_{ik}^{-1}|_{U_{ijk}} g_{jk}|_{U_{ijk}} g_{ij}|_{U_{ijk}} = g_{ijk} for all i, j, k. After replacing \varphi _{ij} by \varphi _{ij}g_{ij}^{-1} we see that \varphi _{ij} gives a descent datum on the objects x_ i over U_ i and the proof is complete.
\square
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