The Stacks project

21.11 Second cohomology and gerbes

Let $p : \mathcal{S} \to \mathcal{C}$ be a gerbe over a site all of whose automorphism groups are commutative. In this situation the first and second cohomology groups of the sheaf of automorphisms (Stacks, Lemma 8.11.8) controls the existence of objects.

The following lemma will be made obsolete by a more complete discussion of this relationship we will add in the future.

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

  1. 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

  2. $H^2(U, \mathcal{G}) = 0$.

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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