Let $B \subset C$ be Noetherian integral domains, and $g \in B$. Thus, $\mathrm{Spec} B \to \mathrm{Spec} B_g$ is an open immersion.
If furthermore $C \subset B_g$, does it follow that $\mathrm{Spec} C \to \mathrm{Spec} B$ is an open immersion?
Let $B \subset C$ be Noetherian integral domains, and $g \in B$. Thus, $\mathrm{Spec} B \to \mathrm{Spec} B_g$ is an open immersion. If furthermore $C \subset B_g$, does it follow that $\mathrm{Spec} C \to \mathrm{Spec} B$ is an open immersion? 


No. $B=k[x,y]$, $g=x$, $C=k[x, x^{1} y]$. 

