I'm reading a book written by Serre and, even though he's one of the best math writer ever, there's a step I don't understand. This may imply that I'm one of the worst math reader ever! :-)
So, we are dealing with a sheaf $S$ on a projective curve $X$ and we can prove that, given a nonzero local section $s$ defined in a neighborhood of a point $P$, there always exists a smaller neighborhood $U$ of $P$ such that $s$ is vanishes on the whole $U\setminus P$. Then the claim is that $$ H^0(X, S) = \bigoplus_{P\in X} \;S_P $$ i.e. the space of global sections coincides with the direct sum of the stalks.
This is intuitively clear for me: from the above we know that the set of points $P$ where a global section $s$ is not trivial is discrete. Moreover, it is closed inside a compact space, thus itself compact. Since a discrete set is compact iff it is finite, we see that $s$ is not trivial only on a finite number of points, and this motivates the presence of $\bigoplus$ in place of $\prod$.
Is my reasoning correct? Is there a better/faster/cleaner way to see it?
Further, is this a general fact, or does it depend on the particular nature (which I didn't describe here) of the sheaf $S$ ? In other words, is it true that if a sheaf $F$ is a skyscrapers sheaf (meaning it's support lies in a finite number of points) on a projective variety, then the above formula for its global sections holds?
One idea would be to understand this via the étalé space associated to the sheaf.
If $\mathcal{F}$ is a sheaf over $X$, the étalé space associated to $\mathcal{F}$ is a topological space which is given as a set by $$\tilde{\mathcal{F}} = \bigsqcup_{p \in X} \mathcal{F}_p$$ with a projection $\pi: \tilde{F} \rightarrow X$ taking each element of $\mathcal{F}_p$ to $p$. This space has the property that for any open set $U \subseteq X$ the sections $\mathcal{F}(U)$ correspond to continuous sections $s: U \rightarrow \tilde{\mathcal F}$ i.e. continuous maps such that $\pi \circ s = id_X$.
In the case of a skyscrapers sheaf with $\mathcal{F}_{p_i} \neq 0$ for $i \in \{ 1, \dotsc, n \}$ then $\tilde{\mathcal F} = \bigsqcup_\limits{i=1}^n \mathcal{F}_{p_i}$ and its global sections $$ s: X \longrightarrow \bigsqcup_\limits{i=1}^n \mathcal{F}_{p_i}, $$ should correspond bijectively with points in $\bigoplus\limits_{i+1}^n \mathcal{F}_{p_i}$.