Consider the following ring extensions: $$A\hookrightarrow\ B\hookrightarrow C$$
where we know that: $A$ is a complete DVR, $B$ and $C$ are Noetherian, Two-dimensional local rings and $B\hookrightarrow C$ is a finite extension. Moreover we know that:
- $A\hookrightarrow C$ is flat and local.
- $A\hookrightarrow B$ is faithfully flat
Can we conclude that also $B\hookrightarrow C$ is flat?
Note: I've seen versions of this statement where assumptions of faithful flatness for $B\hookrightarrow C$ implies the flatness of $A\hookrightarrow B$. But I'm ineterested in a different case
No. Take $A=k[[x^2]]\subset B=k[[x^2,xy,y^2]]\subset C=k[[x,y]]$.