Consider a family of flabby (= flasque) sheaves $(\mathcal F_i)_{i\in I}$ of abelian groups on the topological space $X$.
My question : is their direct sum sheaf $\mathcal F=\oplus _{i\in I} \mathcal F_i$ also flabby?

Here is the difficulty:
Given an open subset $U\subset X$ a section $s\in \Gamma(U,\mathcal F)$ consists in a collection of sections $s_i\in \Gamma(U,\mathcal F_i)$ subject to the condition that for any $x\in U$ there exists a neighbourhood $x\in V\subset U$ on which almost all $s_i\vert V \in \Gamma(V,\mathcal F_i)$ are zero.
Now, every $s_i$ certainly extends to a section $S_i\in \Gamma(X,\mathcal F_i)$ by the flabbiness of $\mathcal F_i$.
The problem is that I see no reason why the collection $(S_i)_{i\in I}$ should be a section in $\Gamma(X,\oplus _{i\in I} \mathcal F_i)$, since I see no reason why every point in $X$ should have a neighbourhood $W$ on which almost all the restrictions $S_i\vert W$ are zero.
Of course any direct sum of flabby sheaves is flabby if the space $X$ is noetherian, since in that case we have $\Gamma(U,\mathcal F) =\oplus_{i\in I} \Gamma(U,\mathcal F_i)$ for all open subsets $U\subset X$.
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...


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 $U\subseteq X$ are the functions $U\to\mathbb F_2$ (not necessarily continuous). This is a flabby sheaf. I claim that the infinite direct sum $\mathcal F^{\oplus\mathbb N}$ of countably many copies of $\mathcal F$ is not flabby.

To see this, let $U=X\setminus\{0\}$, and for $i\in\mathbb N$ let $s_i\colon U\to\mathbb F_2$ 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 $s=(s_i)_{i\in\mathbb N}\in\Gamma(U,\mathcal F^{\oplus\mathbb N})$, 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 $\mathcal F^{\oplus\mathbb N}$ over all of $X$. Indeed, if $s$ extended to a section $\tilde s=(\tilde s_i)_{i\in\mathbb N}$, 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.

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    $\begingroup$ What about finite direct sums? (Or is that a basic sheaf theory fact?) $\endgroup$ – Chris Gerig Jul 19 at 14:05
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    $\begingroup$ @ChrisGerig Flabbyness is preserved for arbitrary direct products and a finite sum of abelian sheaves is the same as their finite product. $\endgroup$ – Chris Jul 19 at 14:24
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    $\begingroup$ Thank you, Alexander: this is a perfect counterexample. I hope it will find its place in some basic book using sheaves, or maybe become an item in the Stacks Project. $\endgroup$ – Georges Elencwajg Jul 19 at 18:08
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    $\begingroup$ @GeorgesElencwajg It will. :-) I am writing a sequel to ams.org/open-math-notes/omn-view-listing?listingId=110823 that will be about homological methods in commutative algebra. The first half or so is about homological algebra per se, and there sheaves appear as a fundamental recurring example. I am putting Alexander's counterexample as a guided exercise. $\endgroup$ – Andrea Ferretti Jul 24 at 13:48
  • $\begingroup$ Dear@Andrea, this is wonderful news! When I wrote about my hope, I actually thought that its realization was not very probable: I'm happy you are showing me that I was unduly pessimistic! I am delighted that you are writing a follow-up to the book I downloaded a few months ago from the AMS site, and which I much appreciated. Thanks a lot for that great document and my best wishes for the future one. $\endgroup$ – Georges Elencwajg Jul 25 at 7:55

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