Exercise 7, § 5

Let $A$ be a ring, $B$ an $A$-algebra; suppose that the $A$-module $B$ is free of finite type. In the following exercises, we denote by $B^*$ the $A$-module $\mathrm{Hom}_A(B, A)$, and we equip it with its natural structure of $B$-module. (a) Let $(e_i)_{i \in I}$ be a basis of the $A$-module $B$, and $(e_i^*)_{i \in I}$ the dual basis of $B^*$. Prove that the element $\mathrm{Tr}_{B/A}$ of $B^*$ (A, III, p. 110) is equal to $\sum_i e_i e_i^*$. (b) Suppose that the $B$-module $B^*$ is free of rank 1; let $\ell$ be an element forming a basis of this module, and let $\delta$ be the element of $B$ such that $\mathrm{Tr}_{B/A} = \delta \ell$. The ideal $\delta B$ of $B$ does not depend on the choice of $\ell$; it is called the different ideal of $B$ over $A$. Prove that $N_{B/A}(\delta)$ is a generator of the discriminant ideal of $B$ over $A$ (A, III, p. 115; calculate the discriminant of a basis $(e_i)$ of $B$ over $A$ using the matrix of the multiplication by $\delta$ in this basis).