Suppose $A_p$ is the stalk of a ring $A$ at a prime ideal $p$.
Consider the (opposite) system of those open immersions $\operatorname{Spec}(A)\leftarrow \operatorname{Spec}(B)$ such that the scheme map $\operatorname{Spec}(k(p))\to \operatorname{Spec}(A)$ factorizes over these $\operatorname{Spec}(B)\to \operatorname{Spec}(A)$. Here, $k(p)=A_p/m_p$ denotes the residue field of $p$. Geometrically, these are smaller and smaller opens around the point $p$ of $\operatorname{Spec}(A)$, I think.
Is it true that $\underset{\rightarrow}{\lim} B=A_p$?