Is a direct sum of flabby sheaves flabby?

Question

Consider a family of flabby (= flasque) sheaves $(mathcal F_i)_{iin I}$ of abelian groups on the topological space $X$.
My question : is their direct sum sheaf  $mathcal F=oplus _{iin I} mathcal F_i$ also flabby?
Here is the difficulty:
Given an open subset $Usubset X$ a section $sin Gamma(U,mathcal F)$ consists in  a collection of sections $s_iin Gamma(U,mathcal F_i)$ subject to the condition that for any $xin U$ there exists a neighbourhood $xin Vsubset U$ on which almost all $s_ivert V in Gamma(V,mathcal F_i)$ are zero.
Now, every $s_i$ certainly extends to a section $S_iin Gamma(X,mathcal F_i)$ by the flabbiness of $mathcal F_i$.
The problem is that I see no reason why the collection $(S_i)_{iin I}$ should be a section in $Gamma(X,oplus _{iin I} mathcal F_i)$, since  I see no reason why every point in $X$ should have a neighbourhood $W$ on which almost all the restrictions $S_ivert W$ are zero.
Of course any direct sum of flabby sheaves is flabby if the space $X$ is noetherian, since in that case we have $Gamma(U,mathcal F) =oplus_{iin I} Gamma(U,mathcal F_i)$ for all open subsets $Usubset X$.
I have only seen the fact that direct sums of flabby sheaves are flabby (correctly) used on noetherian spaces, actually schemes, so that my question originates just from idle curiosity...

Alexander Betts · Answer

No, a direct sum of flabby sheaves need not be flabby.
Take $X={1,1/2,1/3,1/4,dots}cup{0}$ with the subspace topology from $mathbb R$, and let $mathcal F$ be the sheaf whose sections over an open $Usubseteq X$ are the functions $Utomathbb F_2$ (not necessarily continuous). This is a flabby sheaf. I claim that the infinite direct sum $mathcal F^{oplusmathbb N}$ of countably many copies of $mathcal F$ is not flabby.
To see this, let $U=Xsetminus{0}$, and for $iinmathbb N$ let $s_icolon Utomathbb F_2$ denote the function sending $1/i$ to $1$ and all other elements of $U$ to $0$. Thus each $s_i$ is a section of $mathcal F$ over $U$. Observe that $s=(s_i)_{iinmathbb N}inGamma(U,mathcal F^{oplusmathbb N})$, since locally on $U$ all but finitely many of the sections $s_i$ are equal to zero (the topology on $U$ is discrete).
I claim that this section $s$ doesn't extend to a section of $mathcal F^{oplusmathbb N}$ over all of $X$. Indeed, if $s$ extended to a section $tilde s=(tilde s_i)_{iinmathbb N}$, then there would be a neighbourhood of $0$ in $X$ on which all but finitely many of the $tilde s_i$ were equal to $0$. But this would imply that $tilde s_i(1/i)=s_i(1/i)=0$ for all sufficiently large $i$, which is impossible. Thus $s$ does not extend.

Is a direct sum of flabby sheaves flabby?

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