Let $X$ be a scheme, $F$ and $G$ be sheaves in groups on $X$ for the étale topology and $f:F\rightarrow G$ a morphism of group sheaves. Assume that there exists an étale covering $i:U\rightarrow X$ such that the pullback $i^{*}f:i^{*}F\rightarrow i^{*}G$ admits a section, can I lift this section to a section of $f:F\rightarrow G$?
Lifting étale sections
1 Answers
Not in general, no.
(By the way, I would write "descend" rather than "lift"; you are trying to pass from a property holding locally on the cover $U$ to a property over all of $X$.)
To fix ideas, suppose that $G = \mathbb Z/n,$ the constant sheaf attached to a cyclic group of order $n$, that $F$ also has exponent $n$, and that that $F \to G$ is a surjective morphism. Then by definition of surjectivity for sheaves, we may find an etale cover $U$ of $X$ such that the section $1$ of $G$ lifts to a section of $F$ over $U$, and this provides a splitting of $F_{| U } \to G_{| U}$ (since this lift also has order $n$, by our assumption on the exponent of $F$). But the original surjection $F \to G$ need not be split.
E.g. Any non-trivial extension of $\mathbb Z/n$ by $\mathbb Z/n$ as a $\mathbb Z/n$-sheaf will give such a non-split extension, and such extensions are computed by the etale cohomology group $H^1(X,\mathbb Z/n)$, which need not vanish in general.