Lemma 52.5.1. Let $(A, \mathfrak m)$ be a Noetherian local ring.

1. Let $M$ be a finite $A$-module. Then the $A$-module $H^ i_\mathfrak m(M)$ satisfies the descending chain condition for any $i$.

2. Let $U = \mathop{\mathrm{Spec}}(A) \setminus \{ \mathfrak m\}$ be the punctured spectrum of $A$. Let $\mathcal{F}$ be a coherent $\mathcal{O}_ U$-module. Then the $A$-module $H^ i(U, \mathcal{F})$ satisfies the descending chain condition for $i > 0$.

Proof. We will prove part (1) by induction on the dimension of the support of $M$. The statement holds if $M = 0$, thus we may and do assume $M$ is not zero.

Base case of the induction. If $\dim (\text{Supp}(M)) = 0$, then the support of $M$ is $\{ \mathfrak m\}$ and we see that $H^0_\mathfrak m(M) = M$ and $H^ i_\mathfrak m(M) = 0$ for $i > 0$ as is clear from the construction of local cohomology, see Dualizing Complexes, Section 47.9. Since $M$ has finite length (Algebra, Lemma 10.52.8) it has the descending chain condition.

Induction step. Assume $\dim (\text{Supp}(M)) > 0$. By the base case the finite module $H^0_\mathfrak m(M) \subset M$ has the descending chain condition. By Dualizing Complexes, Lemma 47.11.6 we may replace $M$ by $M/H^0_\mathfrak m(M)$. Then $H^0_\mathfrak m(M) = 0$, i.e., $M$ has depth $\geq 1$, see Dualizing Complexes, Lemma 47.11.1. Choose $x \in \mathfrak m$ such that $x : M \to M$ is injective. By Algebra, Lemma 10.63.10 we have $\dim (\text{Supp}(M/xM)) = \dim (\text{Supp}(M)) - 1$ and the induction hypothesis applies. Pick an index $i$ and consider the exact sequence

$H^{i - 1}_\mathfrak m(M/xM) \to H^ i_\mathfrak m(M) \xrightarrow {x} H^ i_\mathfrak m(M)$

coming from the short exact sequence $0 \to M \xrightarrow {x} M \to M/xM \to 0$. It follows that the $x$-torsion $H^ i_\mathfrak m(M)[x]$ is a quotient of a module with the descending chain condition, and hence has the descending chain condition itself. Hence the $\mathfrak m$-torsion submodule $H^ i_\mathfrak m(M)[\mathfrak m]$ has the descending chain condition (and hence is finite dimensional over $A/\mathfrak m$). Thus we conclude that the $\mathfrak m$-power torsion module $H^ i_\mathfrak m(M)$ has the descending chain condition by Dualizing Complexes, Lemma 47.7.7.

Part (2) follows from (1) via Local Cohomology, Lemma 51.8.2. $\square$

Comment #6796 by on

Part (1) can be proven with elementary methods by induction on the dimension of the support of $M$ and using material from Section 47.7. This result should be moved earlier (before the local duality theorem).

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