The Stacks project

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. $\square$

Comments (2)

Comment #6796 by on

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

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