Exercise 9, § 10

Let $A$ be a noetherian local ring, $d$ its dimension, $x = (x_1, \ldots, x_d)$ a maximal sequence of elements of $\mathfrak{m}_A$.

a) Prove that there exists an integer $m_0$ such that for $m \geq m_0$ and $n \geq 0$, one has $(x_1^m \ldots x_d^m)^n \not\in (x_1^{m(n+1)}, \ldots, x_d^{m(n+1)})$ (use exerc. 4 and 7, b)).

b) Suppose that $A$ contains a field of characteristic $p$. Prove that one has $(x_1 \ldots x_d)^n \not\in (x_1^{n+1}, \ldots, x_d^{n+1})$ for every $n \geq 0$, that is to say that $A$ satisfies the condition

(CM) (exerc. 3 of § 2; consider the endomorphism $x \mapsto x^p$ of $A$). Every noetherian local ring containing a field therefore satisfies the condition (CM)$^1$.

c) Suppose in addition that $A$ is regular; let $\rho : A \to B$ be an injective ring homomorphism making $B$ a finitely generated $A$-module. Prove that the $A$-module $\rho(A)$ is a direct factor in $B$ ($cf.$ § 2, exerc. 3); reduce to the case where $B$ is integral, hence local, and apply exerc. 8 of § 4).