The local cohomology modules are Artinian

Question

Let $(R,m,k)$ be Noetherian local ring and $M$ a finitely generated $R$-module. Lemma 3.5.4 of Bruns-Herzog states that

the local cohomology modules $H^i_m(M)$ are Artinian

and that this follows from the structure of $\Gamma_m(E^\bullet(M))$ where $E^\bullet(M)$ is the minimal injective resolution of $M$ and $\Gamma_m(N)=\{x\in N:\exists k\geq0\;\;\; m^kx=0\}$. Now $\Gamma_m(E^\bullet(M))$ looks like

$0\rightarrow E(k)^{\mu_0(m,M)}\rightarrow E(k)^{\mu_1(m,M)}\rightarrow\cdots\rightarrow E(k)^{\mu_i(m,M)}\rightarrow\cdots$

And remember that $H_m^i(-)$ are the derived right functors of $\Gamma_m(-)$.

So my first instinct was to check if in general $E(k)$ was Artinian and it is not.

Could you tell me how to prove that the local cohomology modules are Artinian?

Answer 1

$E(k)$ is artinian. For a proof of this result you can use the same book, proof of Theorem 3.2.13, step (3) and Exercise 3.2.14(b), or Matsumura, Theorem 18.6(v).