Let $(A,m_A)$ and $(B,m_m)$ be two Noetherian local rings, $A \subseteq B$ and $B$ is a finitely generated $A$-algebra.
Step 1: Assume that:
(1) $A$ is regular.
(2) $A \subseteq B$ is flat.
Question 1: Why $B$ is Cohen-Macaulay?
Step 2: Further assume that $m_AB=m_B$, namely, the extension of $m_A$ in $B$ is $m_B$.
Question 2: Why $A \subseteq B$ is unramified? For more details on unramifiedness, see this.
(Then flat+ unramified= etale, and hence $B$ is actually regular, see a and b or c).
Please see this question.
Thank you very much!
Edit: Now I asked the above Question 2 at MO.
Edit 2: The definition of an unramified (local) ring extension can be found here.