Proposition 1

Let $A$ be a noetherian local ring and $M$ a finitely generated $A$-module.

a) The $A$-module $H_A(M)$ is artinian, and zero in degree $> \dim(M)$.

b) Let $p = \operatorname{prof}_A(M)$. We have $H_A^i(M) = 0$ for $i < p$, and $H_A^p(M) \neq 0$ if $M$ is nonzero.

Let us prove a) by induction on $\dim(M)$. The case where $\dim(M) \leq 0$ follows from Example 2 above. Suppose $\dim(M) > 0$ and first take $M$ of the form $A/\mathfrak{p}$, where $\mathfrak{p}$ is a prime ideal of $A$ distinct from $\mathfrak{m}_A$. Let $x$ be an element of $\mathfrak{m}_A - \mathfrak{p}$; we have an exact sequence $0 \to M \xrightarrow{xM} M \longrightarrow M/xM \to 0$, with $\dim(M/xM) = \dim(M) - 1$. Hence we deduce an exact sequence of local cohomology

Every element of $H_A^i(M)$ is annihilated by a power of $\mathfrak{m}_A$; to prove that this module is artinian, it is enough therefore to prove that the socle of $H_A^i(M)$ is of finite dimension over $\kappa_A$ (§ 8, No. 3, Lemma 3). By the induction hypothesis, the kernel $N$ of the homothety of ratio $x$ in $H_A^i(M)$ is artinian; since $x$ belongs to $\mathfrak{m}_A$, the socle of $H_A^i(M)$ identifies with that of $N$, and is therefore of finite dimension. If $i > \dim(M)$, we have $H_A^{i-1}(M/xM) = 0$ by the induction hypothesis, so that the homothety of ratio $x$ is injective in $H_A^i(M)$; since every element of $H_A^i(M)$ is annihilated by a power of $x$, we deduce that $H_A^i(M) = 0$, whence a) in the case considered.

Let us pass to the general case. The $A$-module $M$ admits a composition series $(M_j)_{0 \leq j \leq n}$ such that each quotient $M_j/M_{j+1}$ is isomorphic to $A/\mathfrak{p}_j$, where $\mathfrak{p}_j$ is a prime ideal of $A$ (IV, § 1, No. 4, Theorem 1). Let us prove by induction on $n$ that $M$ satisfies a). The case $n = 0$ is trivial. The exact sequence $0 \to M_1 \to M \to A/\mathfrak{p}_0 \to 0$ yields an exact sequence of local cohomology

the $A$-module $H_A^i(M_1)$ is artinian by the induction hypothesis, and the same is true of $H_A^i(A/\mathfrak{p}_0)$ by the cases already treated; consequently $H_A^i(M)$ is artinian. If $i > \dim(M)$, the modules $M_1$ and $A/\mathfrak{p}_0$ are of dimension $< i$; the modules $H_A^i(M_1)$ and $H_A^i(A/\mathfrak{p}_0)$ are therefore zero by the induction hypothesis and the cases already treated, which implies that $H_A^i(M) = 0$.

Suppose $M$ nonzero, and let us prove b) by induction on the integer $p = \operatorname{prof}(M)$. The case $p = 0$ follows from the definition of depth. Suppose $p > 0$ and choose an element $x$ of $\mathfrak{m}_A$ such that the homothety $x_M$ is injective. As above we obtain a local cohomology exact sequence

We have $\operatorname{prof}(M/xM) = \operatorname{prof}(M) - 1$ (§ 1, No. 4, prop. 7), whence $H_A^{i-1}(M/xM) = 0$ for $i < p$ by the induction hypothesis, which implies as above that $H_A^i(M) = 0$. In particular $H_A^{p-1}(M)$ is zero, so that the homomorphism $H_A^{p-1}(M/xM) \longrightarrow H_A^p(M)$ is injective; thus $H_A^p(M)$ is nonzero by the induction hypothesis.

It can be shown that the module $H_A^{\dim(M)}(M)$ is nonzero when $M$ is nonzero (exerc. 4; cf. No. 3, cor. of th. 2).