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This is a cross-post from MathStackexchange.

We define a flasque sheaf on a site as one whose first Čech cohomology vanishes for every covering of every object of the site. I know this definition is non-standard, but Lei Fu's 'Étale cohomology theory' uses it and shows it to be equivalent to the standard one (i.e, that all higher cohomology groups vanish for every object).

In the proof, he uses the following lemma which I could not prove: If $$0\rightarrow F'\rightarrow F\rightarrow F''\rightarrow 0$$ is an exact sequence and $F'$ and $F$ are flasque, so is $F''$. I am not entirely sure if this is true or if Fu has made a mistake, because I have never seen this claim made anywhere else.

Here is the relevant section from the book:

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As an aside, how does this relate to the 'restriction maps are surjective' definition?

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    $\begingroup$ The Stacks project calls a sheaf $\mathscr F$ totally acyclic if $H^i(\mathbf T/U,\mathscr F) = 0$ for all $i > 0$ and all objects $U$ of the topos $\mathbf T$, and claims without details that (1) it is not enough to check on representable objects $U$ (coming from the site), and (2) this clashes with the 'restrictions are surjective' definition. See Tag 079X. There is no remark on whether $H^1$ suffices. $\endgroup$
    – R. van Dobben de Bruyn
    Commented Apr 19, 2022 at 9:27
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    $\begingroup$ I am just clarifying: are you asking whether this is true for every site, or just for the etale site of a scheme (the only site where Lei Fu claims that it is true)? $\endgroup$
    – Jason Starr
    Commented Apr 19, 2022 at 10:56
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    $\begingroup$ Exercise: on the \'etale site of a scheme, there are no "flasque" sheaves of abelian groups apart from the zero sheaf, if flasque is defined by "surjective restriction maps". $\endgroup$
    – Johan
    Commented Apr 19, 2022 at 12:30
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    $\begingroup$ @Johan I see. For any $U$, take the covering $U\sqcup U\rightarrow U$. $\endgroup$
    – Jehu314
    Commented Apr 19, 2022 at 13:03
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    $\begingroup$ @Jehu314 If you have a sheaf $F$ such that $\textrm{Ext}^1 (E, F) = 0$ for a sufficiently large class of sheaves $E$, then $F$ will have the property that "restrictions along monomorphisms are surjective": given $U \subseteq V$, we can form the short exact sequence of sheaves $0 \to \mathbb{Z} U \to \mathbb{Z} V \to \mathbb{Z} U / \mathbb{Z} V \to 0$ and apply $\textrm{Ext}^*(- , F)$ to get an exact sequence $F (V) \to F (U) \to \textrm{Ext}^1 (\mathbb{Z} V / \mathbb{Z} U, F)$. The point is that in a general site, morphisms in the site are not always monomorphisms. $\endgroup$
    – Zhen Lin
    Commented Apr 20, 2022 at 15:07

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