I am searching for a locally constant sheaf which is not constant. Locally constant means that there exists an open covering of the total space such that the sheaf restricted to each open set in the cover is isomorphic to a constant sheaf. But the sheaf should not be constant throughout X. Does there exist such an example. Please help.
Sheaf Theory – Locally Constant Sheaf but Not Constant
sheaf-theory
Best Answer
If $X$ is locally connected, locally constant sheaves are (up to isomorphism) exactly the sheaves of sections of covering spaces $\pi:Y\to X$.
Such a locally constant sheaf is a constant sheaf if and only the covering $\pi$ is trivial.
So any non trivial covering will give you a non-constant but locally constant sheaf.
The simplest example is the sheaf of sections of the two sheeted non trivial covering $\mathbb C^*\to \mathbb C^*:z\mapsto z^2$ or its restriction to the unit circle $S^1\to S^1: e^{i\theta}\mapsto e^{2i\theta}$