I'm trying to wrap my head around the fact that if I take a global section $ f \in \Gamma(X, \mathcal{O}_X) $. Then the support $ Supp (s) = \{ p \in X : s_p \neq 0 \} $ is closed, but the locus $ X_s = \{ p \in X : s_p \notin \mathfrak{m}_p\mathcal{O}_p \} $ is open.
Any help in understanding this difference would be appreciated. Also to help with this
I tried to calculate $ Supp(s) $ and $ X_s $, for $ X = Spec (k[x,y]) $ and $ s = y-x^2 $. But had
no idea on how to calculate the germ at a point $ p \in X $. Would also appreciate some help with this.
Best Answer
Here is the simplest example showing the difference between $supp(s)$ and $X_s$.
Let $X=\mathbb R$ and $\mathcal F=\mathcal C$, the sheaf of real valued continuous functions on $\mathbb R$.
Consider the identity function $s\in \mathcal C(\mathbb R)$ defined by $s(t)=t$ for $t\in \mathbb R$.
Then $X_s=\mathbb R\setminus\{0\}$, which is open, and $supp(s)=\mathbb R$, which is closed.
Remark
Does this example seem childish? Of course it is! I think the duty of a teacher is to make as large a chunk of mathematics as possible seem childish, especially when introducing new concepts.