# Is a direct sum of flabby sheaves flabby?

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.

• What about finite direct sums? (Or is that a basic sheaf theory fact?) – Chris Gerig Jul 19 at 14:05
• @ChrisGerig Flabbyness is preserved for arbitrary direct products and a finite sum of abelian sheaves is the same as their finite product. – Chris Jul 19 at 14:24
• 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. – Georges Elencwajg Jul 19 at 18:08
• @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. – Andrea Ferretti Jul 24 at 13:48
• 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. – Georges Elencwajg Jul 25 at 7:55