Exercise 13, § 1

Let $A$ be a ring, $J$ an ideal of $A$, $M$ an $A$-module. For every integer $n \geq 0$, denote by $\mathrm{prof}_n(J;M)$ the upper bound (in $\overline{\mathbf{N}}$) of the lengths of the $M[X_1,\ldots,X_n]$-regular sequences of elements of $JA[X_1,\ldots,X_n]$. a) Now prove that the sequence $(\mathrm{prof}_n(J;M))_{n \geq 0}$ is increasing; denote by $\mathrm{prof}_\infty(J;M)$ its limit (in $\overline{\mathbf{N}}$). Prove that one has $\mathrm{prof}_\infty(J;M) \leq \mathrm{prof}_A(J;M)$.

EXERCISES

b) When the ideal $J$ is of finite type, prove the equality $\mathrm{prof}_{\infty}(J; M) = \mathrm{prof}_A(J; M)$ (reason as in the proof of th. 2 of no. 4, using exerc. 12).

c) In the general case, show that $\mathrm{prof}_{\infty}(J; M)$ is the upper bound of the numbers $\mathrm{prof}_A(J'; M)$, where $J'$ runs through the set of ideals of finite type contained in $J$.

d) Let $J'$ be an ideal of $A$ such that $V(J') = V(J)$. Prove that one has $\mathrm{prof}_{\infty}(J'; M) = \mathrm{prof}_{\infty}(J; M)$.

e) Let $A$ be a valuation ring of height 1, non noetherian, $m$ its maximal ideal. Prove that one has $\mathrm{prof}_{\infty}(m; A) = 1$ and $\mathrm{prof}_A(m; A) \geqslant 2$ (cf. exerc. 2).