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This is my first post on - please excuse me if I have overseen some relevant part of the FAQ or this question among those already answered or if I do misbehave in any other way, it maybe takes a bit of use to understand how to find out whether one can ask a question or not.

I am supposed to teach myself the content "Sheaves, Cohomology and the de Rham Theorem" using Warner's Foundations of Differentiable Manifolds and Lie Groups. I have an issue with a very early supposedly easy problem and this suggest that I do not get any grip on the objects I am dealing with. Could somebody maybe give me a hint or a point of view which stays in Warner's framework (no ringed spaces a.s.o.)? Before I delve into the questions, I fix a bit of vocabulary (as this might vary from author to author):

  • $M$ is a manifold
  • $K$ is a principal ideal domain
  • A sheaf $\mathcal{S}$ of $K$-modules over $M$ is a tuple $(\mathcal{S},\pi)$ where $\mathcal{S}$ is a topological space and $\pi:\mathcal{S}\to M$ is a local homeomorphism such that for all $m\in M$ the preimage $\pi^{-1}(m)$ is a $K$-module and the composition laws (on the modules in) $\mathcal{S}$ are continuous.

Now for the question: Let $(\mathcal{S},\pi)$ be a sheaf of $K$-modules over a manifold $M$. Let $U\subset M$ open and $f:U\to\mathcal{S}$ be a section of $\mathcal{S}$ over $U$, i.e. $f$ is continuous and $\pi\circ f=\operatorname{id}_{U}$. Assume $m\in U$ such that $f(m)=0$ in $\pi^{-1}(m)$. Then there exists a neighbourhood $m\in V\subset U$ such that $f$ vanishes on all of $V$, i.e. $f(v)=0\in\pi^{-1}(v)$.

I assume that this must be true as of the following: "[...] if sections $f$ and $g$ agree at $m\in M$, then they must agree on a neighbourhood of $m$".

As I had no idea how to answer this question, I decided to first work on the first problem in the exercise section, i.e. that the null-section $g:M\to\mathcal{S}$ defined by $g(m)=0\in\pi^{-1}(m)$ is continuous. I haven't even managed to prove this. That's why I am asking for help. I have tried to prove it pretty directly, i.e. I chose a neighbourhood $g(m)=0\in W\subseteq\mathcal{S}$ such that $\pi\big|_{W}$ is a homeomorphism and then started looking for a neigbourhood $m\in U\subseteq M$ which satisfies $g(U)\subseteq W$. I used some neighbourhood $m\in V\subseteq\pi(W)$ and $\pi\circ g\big|_{V}=\operatorname{id}_{V}$ to deduce that $g(V)\subseteq \pi^{-1}(\pi(W))$. The latter simply is: $$ \pi^{-1}(\pi(W))=\bigcup_{w\in W}(\{0\}\cup\pi^{-1}(\pi(w))\setminus\{0\}) $$ which is what I did stare at but this did not help. I feel that I really do not get a grip on these objects, so that is why I am turning to you.

  • Maybe somebody could give me a hint on what to do
  • More usefully maybe somebody could give me a hint on how to interprete these things in the context of Warner's definition.
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up vote 2 down vote accepted

You can cosider such a sheaf as a collection of modules bunched together in a continuous way, 'over the points of $M$'. Note that the fibres $\pi^{-1}(m)$ are discrete because $\pi$ is a local homeomorphism, hence the topology inherited to the modules is discrete.

Then, try to prove that if $f:X\to Y$ continuous and is a map above the local homeomorphisms $\xi:X\to Z$ and $\eta:Y\to Z$ (that is, $\eta\circ f=\xi$), then $f$ itself is a local homeomorphism, too.

As the module operations are continuous, we also have that the map $\mathcal S\to \mathcal S$ mapping $s_m\mapsto (s_m-s_m)$ i.e. $s_m\mapsto 0_m$ over $m\in M$ is continuous, so, by the above it follows that the subset $\{ 0_m\mid m\in M\}$ is open, and also that the corresponding null section $m\mapsto 0_m$ is continuous.

About the case when sections $f,g$ coincide at a point: $f(m)=g(m)$, use the fact that $\pi$ is a local homeomorphism, that is the sections are local inverses of $\pi$, i.e. there is a neighborhood $U\subset \mathcal S$ of $f(m)=g(m)$, such that $\pi|_U:U\to \pi(U)$ is a homeomorphism, so that $$f|_{\pi(U)}=(\pi|_U)^{-1}=g|_{\pi(U)}. $$

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Thank you so much. I did not realize that the fibres inherit the discrete topology, which is an important piece of information. This also makes it a lot easier to imagine a sheaf. I was always bewildered by the fact that I have very little information on the topology of these bunched together modules. It turns out that I have a lot! I was looking at cases where discreteness was true when thinking about a proof and I expect this to be very helpful. The fact that 'as a germ' the inversion of $\pi$ is unique is very helpful of course. – M. Luethi Nov 11 '12 at 22:02
I did go through everything discussed as above and it seems that the discreteness of the fibers is indeed what I needed in order to match what I hoped sheaves looked like and what they indeed look like. I hope now I can finally get started properly. Thank you once again. – M. Luethi Nov 12 '12 at 9:04

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