Exercise 3, § 2

a) Let $R$ be a Noetherian, integral and integrally closed Q-algebra, $A$ a ring and $\rho : R \to A$ an injective homomorphism, making $A$ a finite $R$-algebra. Prove that the sub-$R$-module $\rho(R)$ is a direct factor of $A$ (reduce to Cor. 3 of Prop. 8). b) Let $A$ be a Noetherian local Q-algebra. Prove that $A$ satisfies the following property («monomial condition»): (CM) for every maximal regular sequence $(x_1, \ldots, x_d)$ of elements of $\mathfrak{m}_A$ and every integer $p$, the element $(x_1 \ldots x_d)^p$ does not belong to the ideal $(x_1^{p+1}, \ldots, x_d^{p+1})$. (Reduce to the case where $A$ is complete, hence admits a coefficient field $K$, and apply a) to a homomorphism $\rho : K[[X_1, \ldots, X_d]] \to A$ such that $\rho(X_i) = x_i$.)