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...



Is a direct sum of flabby sheaves flabby?
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.


