Exercise 13, § 3

Let $A$ be a noetherian local ring. We suppose the following condition¹ is satisfied:

(GM) for every noetherian local $A$-algebra $B$, there exists a $B$-module $M$ such that $m_B M \neq M$ and $\mathrm{prof}_B(M) = \dim(B)$.

(a) Let $P$ be a complex of flat $A$-modules, of finite length $\ell$, such that $\mathrm{Supp}(H(P)) = \{ m_A \}$. Prove that $\ell \geq \dim(A)$ (use exerc. 6 of § 1)².

(b) Let $B$ be a noetherian ring, $\rho : A \to B$ a homomorphism, $^a\rho : \mathrm{Spec}(B) \to \mathrm{Spec}(A)$ the associated mapping, $M$ a finitely generated $A$-module. Prove that the codimension of $^a\rho^{-1}(\mathrm{Supp}(M))$ in $\mathrm{Spec}(B)$ is $\leq \mathrm{dp}_A(M)$ (apply a) to the complex $L \otimes_A B_q$, where $q$ is the minimal prime ideal of a component of $^a\rho^{-1}(\mathrm{Supp}(M))$ and $L$ a free resolution of $M$).

(c) Let $M, N$ be finitely generated $A$-modules, with $M \neq 0$. Prove the inequality

(intersection theorem: reason by induction on the integer $d = \dim(M \otimes_A N)$. If $d = 0$, apply b) to the ring $B = A / \mathrm{Ann}(M)$; in the general case consider an element of $A$ which is secant for $N$ and for $M \otimes_A N$).

(d) For there to exist a finite length $A$-module of finite projective dimension, it is necessary and sufficient that $A$ be a Macaulay ring.