If I have a sheaf $\mathcal{F}$ in a topological space when can I say that it's isomorphic to some constant sheaf? This is when can I say that exists an isomorphism from $\mathcal{F}$ to some constant sheaf?
When is a sheaf isomorphic to a constant sheaf?
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sheaf-theory
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0It will help if you can say more about the situation you have in mind. For example, if you know that the sheaf is locally constant, $\mathcal{F}$ is constant if you the action of $\pi_1(X,x)$ on $\mathcal{F}_x$ is trivial. Of course, if $\mathcal{F}$ is not locally constant, this method does not apply... – 2017-02-13
1 Answers
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Well, a necessary condition is that $\mathcal{F}_x$ is constant for each $x$, but you probably know that it is not enough. The traditionnal way to prove that $\mathcal{F}$ is constant is to construct a morphism of sheaves $$\mathcal{F} \to M$$ (here $M$ is the constant sheaf associated to an abelian group $M$) and to prove that, on the stalks, this map gives you isomosphisms $$\mathcal{F}_x \simeq M.$$ But it is important to notice that you need a global morphism.
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0Yes but, aside from constructing the morphism , exists any condition on the sheaf to guarantee the existence of the isomorphism. – 2017-02-13