Let $A\subset B$ be an injective homomorphism of noetherian integral domains s.t. $B$ is finitely generated as $A$-module and $A$ normal.
Suppose that the normalization $\bar B$ of $B$ is finitely generated as $A$-module. And call $F_A\subset F_B$ the induced extension of fraction fields.
(i) Prove that if $\Omega_{B/A}=0$ then $B$ is normal.
(ii) Prove that if there exists $0\neq s\in A$ such that $\Omega_{[s^{-1}]B/[s^{-1}]A}=0$ iff $F_A\subseteq F_B$ is a separable field extension.
(iii) Provide an example where $\Omega_{B/A}\neq 0$, but $B$ is normal.
(iv) Prove that there exists $0\neq s\in A$ s.t. $B[s^{-1}]=\bar B[s^{-1}]$.
I've tried in this way:
(i) I've considered the conormal sequence, with $B\cong A[X_1, ..., X_n]/(f_1,..., f_m)$, then i've got that, called $I=(f_1, ..., f_m)$, $$I/I^2 \rightarrow \bigoplus_{i=1}^{n} Bdx_i\cong B\otimes_A \Omega_{A[X_1,..., X_n]/A}$$ then map is surjective. My idea was to obtain some information about maximal ideals.
(ii) Done
(iii) I've considered $R=\frac {k[X,Y]}{(Y^2-X(X-1)(X+1))}$ and $A=k[X]$. In fact $R$ is normal since $X(X-1)(X+1)$ has no multiple roots, and $A$ UFD so it's normal. $$\Omega_{R/A}\cong \frac{Rdx\oplus Rdy}{(2ydy+(3x^2-1)dx)R}\neq 0$$
(iv) No idea at all.