Exercise 9, § 5

Let $A$ be a ring, $n$ an integer, $S$ the algebra $A[X_1, \ldots, X_n]$ (resp. $A[[X_1, \ldots, X_n]]$), $a$ an ideal of $S$ generated by a sequence $P = (P_1, \ldots, P_n)$ completely transversal for $S$. Put $B = S/a$, and denote by $\pi$ the canonical homomorphism of $S$ onto $S/a$. Suppose that the $A$-module $B$ is free of finite type. We propose to prove that the $B$-module $B^*$ is free of rank 1 and that the different ideal of $B$ over $A$ is generated by $\pi(\det(\frac{\partial P_i}{\partial X_j}))$. (a) Put $C = S \otimes_A B$ and $\xi_i = X_i \otimes 1 - 1 \otimes \pi(X_i)$ for $1 \leq i \leq n$. Denote by $\varphi : C \to B$ the homomorphism of $A$-algebras such that $\varphi(P \otimes b) = \pi(P)b$. Prove that the kernel of $\varphi$ is generated by the sequence $\xi = (\xi_1, \ldots, \xi_n)$, and that this sequence is completely transversal for $C$. (b) Let $(c_{ij}) \in M_n(C)$ be a matrix satisfying $P_i \otimes 1 = \sum_j c_{ij} \xi_j$. Prove the equality $\pi(\frac{\partial P_i}{\partial X_j}) = \varphi(c_{ij})$. (c) The matrix $(c_{ij})$ defines a morphism of complexes $K_\bullet(P, C) \to K_\bullet(\xi, C)$ (A, X, p. 151), whence a morphism of complexes of $S$-modules $u : K_\bullet(P, S) \to K_\bullet(\xi, C)$. The homomorphism $u_n : S \to C$ maps $1_S$ onto the element $d = \det(c_{ij})$ of $C$. (d) Let $v : \mathrm{Hom}_S(C, S) \otimes_C B \to B$ be the $B$-linear homomorphism such that $v(f \otimes 1) = \pi(f(d))$; prove that $v$ is an isomorphism (identify $v$ with $H^n(\mathrm{Hom}_S(u, 1_S))$), and observe that each of the complexes $K_\bullet(P, S)$ and $K_\bullet(\xi, C)$ defines a resolution of $B$ by free $S$-modules of finite type. (e) Consider the composite mapping $t : \mathrm{Hom}_A(B, A) \to \mathrm{Hom}_S(C, S) \to B$, where the first arrow is obtained by extension of scalars and the second is deduced from $v$. Prove that $t$ is $B$-linear and bijective (identify $\mathrm{Hom}_S(C, S)$ with $\mathrm{Hom}_A(B, A) \otimes_B C$). (f) Prove the formula $t(\mathrm{Tr}_{B/A}) = \varphi(d)$ (express $d$ and $\mathrm{Tr}_{B/A}$ using a basis of $B$ and the dual basis of $B^*$, cf. exerc. 7, a)).

g) Deduce from b), e) and f) that the different ideal of B over A is generated by $\pi(\det(\frac{\partial P_i}{\partial X_j}))$.