Exercise 14, § 3

Let $A$ be a noetherian local ring satisfying condition (GM) (exerc. 13), $M$ an $A$-module of finite type and of finite projective dimension.

a) Prove the inequalities

If moreover the module $M$ is perfect (exerc. 4), one has $\dim(A) = \mathrm{dp}_A(M) + \dim(M)$ (use exerc. 15 of § 1).

b) Let $p \in \mathrm{Supp}(M)$; if the $A_p$-module $M_p$ is Macaulay, the ring $A_p$ is a Macaulay ring. This is the case in particular if $p$ is a minimal prime ideal of $\mathrm{Supp}(M)$.

¹ This assumption is always satisfied when the ring $A$ contains a field, cf. M. HOCHSTER, Topics in the homological theory of modules over commutative rings, Amer. Math. Soc., Providence (1975).

² When $P$ is of finite type, this result is true without the assumption (GM), cf. P. ROBERTS, Le théorème d’intersection, C. R. Acad. Sci. Paris, sér. I, t. 304, 177-180 (1987). Consequently the results of exerc. 13 and 14 are true for every noetherian local ring $A$.

d) Prove that every M-regular sequence is A-regular. (Prove that every element $a$ of $A$ whose annihilator $a$ is not zero is not M-regular; for this, reduce by localization to the case where $\operatorname{Supp}(M) \cap \operatorname{Supp}(a) = \{m_A\}$; by applying exerc. 13 a) to the complex $L \otimes_A A/p$, where $L$ is a free resolution of $M$ and $p$ an associated prime ideal of $a$, prove the inequalities $\operatorname{prof}(A) \leq \dim(A/p) \leq \operatorname{dp}_A(M)$ and deduce $\operatorname{prof}(M) = 0 .$)

e) In order that the ring $A$ be integral, it is necessary and sufficient that it contain a prime ideal of projective dimension $< +\infty$ (if $\operatorname{dp}_A(p) < +\infty$, show that $A$ identifies with a subring of the regular ring $A_p$).