Let $A$ be a Noetherian normal (therefore expecially integral) local ring with unique maximal ideal $\frak{m}$. Let $K$ be it's fraction field, $L$ a finite separable finite field extension of $K$, and $B$ the integral closure of $A$ in $L$. Let consider the associated map $f: X \to Y$ of schemes $X=\operatorname{Spec}(B)$ and $Y=\operatorname{Spec}(A)$, which is well known to be finite.

Question: What is known about it's ramification locus (=$x \in X$ with $(\Omega_{X/Y})_x =0$) (except that it must be open, which holds under much weaker assumptions)?

Background/motivation: Milne's example 4.10 (p 37) in his book ( not the online script!) on Etale Cohomology. There one considered a slightly similar situation: $A$ is normal local as above, $B$ the integral closure of $A$ in sepatable closure $K_s$, and $\mathfrak{n} \subset B$ any maximal ideal in $B$ over $\mathfrak{n} $.

One considers the decomposition group $D_n:= \{g \in G \ | \ g(\mathfrak{n})= \mathfrak{n} \}$ of $\mathfrak{n} $ sitting inside the Galois group $G$ of $K_s/K $ and $B^D$ the integral closure of $A$ in fixed field $K_s^D$.

The question is why $B^D$ is unramified over $A$ at $\mathfrak{n} $, ie why the induced map of local rings $A_{\mathfrak{m} } \to B^D_{\mathfrak{n} }$ is unramified?