Exercise 8, § 4

Let $A$ be a regular local ring, $x = (x_1, \ldots, x_d)$ a system of coordinates of $A$, $\rho : A \to B$ an injective ring homomorphism making $B$ a finitely generated $A$-module. Prove that the following conditions are equivalent:

(i) for every integer $p$, the element $\rho(x_1 \ldots x_d)^p$ does not belong to the ideal of $B$ generated by $\rho(x_1^{p+1}), \ldots, \rho(x_d^{p+1})$; (ii) the $A$-module $\rho(A)$ is a direct factor of $B$.

These conditions are satisfied if $A$ is a $\mathbf{Q}$-algebra (exercise 3 of § 2).

(To prove (ii) $\Rightarrow$ (i), adapt the proof of loc. cit. For the opposite implication, reduce to the case where $A$ is complete, and set for $p \geqslant 0$ $A_p = A/(x_1^{p+1}, \ldots, x_d^{p+1})$ and $B_p = B/(x_1^{p+1}, \ldots, x_d^{p+1})B$. Observe that the class of $(x_1 \ldots x_d)^p$ generates the socle of $A_p$, and deduce that the homomorphism $\rho_p : A_p \to B_p$ induced by $\rho$ is injective. Prove that $\rho_p$ admits an $A_p$-linear retraction. Conclude with the aid of E, III, p. 60, example II.)