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?



If $B \subset C \subset B_g$, is $\mathrm{Spec} C \to \mathrm{Spec} B$ necessarily an open immersion?
No. $B=k[x,y]$, $g=x$, $C=k[x, x^{-1} y]$.


