18
$\begingroup$

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...

Improve this question
$\endgroup$
1
  • $\begingroup$ An example can be found in Bredon's Sheaf Theory book: Example 5.11. $\endgroup$ Commented Dec 31, 2024 at 10:17

1 Answer 1

Reset to default
21
$\begingroup$

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.

Improve this answer
$\endgroup$
5
  • 1
    $\begingroup$ What about finite direct sums? (Or is that a basic sheaf theory fact?) $\endgroup$ Commented Jul 19, 2020 at 14:05
  • 6
    $\begingroup$ @ChrisGerig Flabbyness is preserved for arbitrary direct products and a finite sum of abelian sheaves is the same as their finite product. $\endgroup$ Commented Jul 19, 2020 at 14:24
  • 1
    $\begingroup$ 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. $\endgroup$ Commented Jul 19, 2020 at 18:08
  • 1
    $\begingroup$ @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. $\endgroup$ Commented Jul 24, 2020 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. $\endgroup$ Commented Jul 25, 2020 at 7:55

You must log in to answer this question.

Start asking to get answers

Find the answer to your question by asking.

Ask question

Explore related questions

See similar questions with these tags.