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) I know that if the extension is separable iff $ \Omega_{F_B/F_A}\cong 0$ but i really don’t know how to get some information about $ \Omega_{B/A}$ .

(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.