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I am trying to understand local systems and the first step is to understand constant sheaves. I am confused, because I see different definitions used. For symplicity, I will ask about sheaves over $X$ with values in $\mathbb C$.

  1. The constant presheaf is the sheaf $U \mapsto \mathbb C$ with restriction maps the identity. Its sheafification is called the constant sheaf.
  2. A sheaf is called a constant sheaf, if it is isomorphic to the constant sheaf
    (See on stacksproject)
  3. The sheaf of locally constant functions is called a constant sheaf
  4. The constant sheaf is a sheaf on X whose stalks are all equal to $\mathbb C$.
    (Wikipedia)

Here are my questions:

  1. First of all, 4. does not seem right to me, as it would imply that all line bundles are constant sheafs. Did I understand this correctly?

  2. Is the constant presheaf already a sheaf on reasonable spaces, for example on the interval?

  3. Are $2$ and $3$ equivalent?

  4. And most importantly, how can I show that a sheaf is a constant sheaf? My approach so far was to construct an isomorphism of sheaves to the sheaf whose restriction maps are all the identity. But as I know now, this was not an isomorphism of sheaves, only of pre-sheaves.

I am rather new here, would it be better to ask the questions separately?

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  • 1
    $\begingroup$ Line bundles are sheaves of $\mathcal{O}_X$-modules while local systems will just be sheaves of $\underline{\mathbb{C}}$-modules. So line bundles are not local systems. The constant presheaf is never a sheaf, because a sheaf sends the empty set to $0$. $\endgroup$ – user45878 Jul 4 at 12:41
  • $\begingroup$ Let $X$ be a topological space, then the constant sheaf $\underline{\mathbb{C}}$ on $X$ can be interpreted as the sheaf $U \mapsto \hom_{\text{cont}}(X, \mathbb{C})$, where we consider $\mathbb{C}$ with the discrete topology (so continuous maps are the same thing as locally constant maps). So yes in this sense 2 and 3 are equivalent. $\endgroup$ – user45878 Jul 4 at 12:44
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  1. For sheaves of $\mathcal{O}_X$-modules, there are two different notions stalks and fibers, which should not be mixed up. Whereas a line bundle has fiber isomorphic to $\mathcal{C}$, its stalks are in fact much bigger (isomorphic to the local rings of $\mathcal{O}_X$). So no, line bundle are not constant sheaves and definition 4. is right.

  2. No, the constant presheaf is not a sheaf on an interval. In fact, if $X=[0,1]$ and $U=(0.1,0.2)\cup (0.8,0.9)$, the constant presheaf $\mathcal{P}$ has obviously $\mathcal{P}(U)=\mathbb{C}$ whereas its sheafification has $\mathcal{P}^+(U)=\mathbb{C}^2$ as you can easily see with definition 3. Actually, the constant presheaf is a sheaf iff the space is irreducible. (So this may holds for schemes, but not for Hausdorff topological spaces unless it is trivial).

  3. Weird question, definition 3. only gives an example of a constant sheaf.

  4. This is often a hard problem. It might uses some advanced notions (monodromy...). At least, if you can construct a global section of your sheaf $\mathcal{F}$, you then have a morphism from the constant sheaf to $\mathcal{F}$ and you can now check that this morphism is an isomorphism on stalks. But constructing a global section might be very hard...

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  • $\begingroup$ Thank you,this clears some things up. But still, what is the definition of the constant sheaf? I see you also mix this up. To 4.: If $\mathcal F(X) = \mathbb C$, then I see how to construct a morphism from the constant presheaf to $\mathcal F$ (map to global section and then restrict), but how is it done with the constant sheaf? I know that it exists because of the universal property of sheafication though. Wouldn't it be easier to just work on the basis of presheafs? The stalks are still the same in the end $\endgroup$ – tralala Jul 4 at 15:35
  • $\begingroup$ What is wrong with the definitions you gave (1 and 2) ? But in fact, we never speak of the constant sheaf (in fact, we never speak of the vector space of dimension $n$). Yes, as you said, you use the universal property of sheafification. It is not easier to work on a basis so that we can stay in the category of sheaves. $\endgroup$ – Roland Jul 4 at 21:11
  • $\begingroup$ Nothing is wrong with the definition, I was just confused about the different definitions and if they are equivalent. As you said, distinguishing THE constant sheaf and A constant sheaf seems weird. What did you mean with you last sentence? $\endgroup$ – tralala Jul 4 at 22:31
  • $\begingroup$ I mean, except in category theory, we are not very much interested in presheaves. If we can stay in the category of sheaves, then this is better. A global section is just a map of sheaves $s:\underline{\mathbb{C}}\to\mathcal{F}$ where the source is the constant sheaf. The stalks of the source are just $\mathbb{C}$, so you just have to check if $s$ induces isomorphisms $s:\mathbb{C}\to\mathcal{F}_x$. $\endgroup$ – Roland Jul 5 at 6:45
  • $\begingroup$ Sorry, for the late reply, but would you pls still help me? I dont't see how to define the morhphism $\underline {\mathbb C } \to \mathcal F$. As I said, I can see it for the constant presheaf, using that the restriction maps in the constant presheaf are the identity (actually, I think I only use surjectivity for this). But how is it done for the constant sheaf? $\endgroup$ – tralala Jul 13 at 16:39

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