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. Proof of (1). Let $A^\wedge $ be the completion of $A$. Since $H^ i_\mathfrak m(M)$ is $\mathfrak m$-power torsion, we see that $H^ i_\mathfrak m(M) = H^ i_\mathfrak m(M) \otimes _ A A^\wedge $. Moreover, we have $H^ i_\mathfrak m(M) \otimes _ A A^\wedge = H^ i_{\mathfrak mA^\wedge }(M \otimes _ A A^\wedge )$ by Dualizing Complexes, Lemma 47.9.3. Thus

\[ H^ i_\mathfrak m(M) = H^ i_{\mathfrak mA^\wedge }(M \otimes _ A A^\wedge ) \]

and $A$-submodules of the left hand side are the same thing as $A^\wedge $-submodules of the right hand side. Thus we reduce to the case discussed in the next paragraph.

Assume $A$ is complete. Then $A$ has a normalized dualizing complex $\omega _ A^\bullet $ (Dualizing Complexes, Lemma 47.22.4). By the local duality theorem (Dualizing Complexes, Lemma 47.18.4) we find an isomorphism

\[ \mathop{\mathrm{Hom}}\nolimits _ A(H^ i_\mathfrak m(M), E) = \text{Ext}^{-i}_ A(M, \omega _ A^\bullet )^\wedge \]

where $E$ is an injective hull of the residue field of $A$. The module $\text{Ext}^{-i}_ A(M, \omega _ A^\bullet )$ on the right hand side is a finite $A$-module by Dualizing Complexes, Lemma 47.15.3. Since $A$ is complete, the completion isn't necessary. Thus $H^ i_\mathfrak m(M)$ has the descending chain condition by Matlis duality, see Dualizing Complexes, Proposition 47.7.8 and its addendum Remark 47.7.9.

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

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