I have to solve exercise 2.3 by Shafarevich - Basic algebraic geometry, vol. II.
Let $X$ be a topological space, $M$ an Abelian group and $\mathfrak{F}(U)$ the quotient group of all locally constant functions on $U$ with values in $M$ by the constant functions; for $U \subset V$, $\rho_U^{V}$ is restriction. Prove that $\mathfrak{F}$ is a presheaf and determine its sheafication.
It's easy to prove that $\mathfrak{F}$ is a presheaf, because of the properties of restrictions and of locally constant functions. Now, I know that sheafication is the family of germs $\{ u_x \in \mathfrak{F}_x\}_{x \in U}$ such that for all $x \in U$, there exists a neighbourhood $x \in W \subset U$ and an element $w \in \mathfrak{F}(w)$ such that $u_y=\rho_y^{W}(w)$ for all $y \in W$. At a first glance, I think it's the set of locally constant $M-$valued functions, but I'm not sure of this and I can't give a proof of t his fact.
Let $\mathscr{C} (U)$ be the set of constant functions $U \to M$ (where $U \subseteq X$ is open and $M$ is considered to have the discrete topology) and let $\mathscr{L} (U)$ be the set of locally constant functions $U \to M$. It is not hard to see that $\mathscr{C}$ and $\mathscr{L}$ are presheaves on $X$. By definition, we have the following short exact sequence of presheaves, $$0 \to \mathscr{C} \to \mathscr{L} \to \mathscr{F} \to 0$$ so the induced sequence of sheaves is also exact: $$0 \to \tilde{\mathscr{C}} \to \tilde{\mathscr{L}} \to \tilde{\mathscr{F}} \to 0$$ Hence, $\tilde{\mathscr{F}} = 0$ if and only if $\tilde{\mathscr{C}} \to \tilde{\mathscr{L}}$ is an isomorphism. I leave it to you to show that $\tilde{\mathscr{C}} \to \tilde{\mathscr{L}}$ is indeed an isomorphism.