Stalks of ringed space

Question

Let $X$ be a locall ringed space (more narrowly a scheme, if you like) and $A=\Gamma(X,\mathcal{O}_X)$ its ring of global sections. Given a point $x\in X$, is there a prime ideal $p$ of $A$ such that $A_p$ is the stalk $\mathcal{O}_{X,x}$? I have been asking myself this for a while, but I couldn't figure it out. I know it is true by Definition for affine schemes.

Answer 1

I think $\mathbb P^1(\mathbb C)$ gives you a simple counter-example. Its ring of global sections is simply $\mathbb C$ whereas its local rings are of the form $\mathbb C[t]_{(t-\alpha)}$.