52.5 Mittag-Leffler conditions

When taking local cohomology with respect to the maximal ideal of a local Noetherian ring, we often get the Mittag-Leffler condition for free. This implies the same thing is true for higher cohomology groups of an inverse system of coherent sheaves with surjective transition maps on the puncture spectrum.

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. $\square$

Lemma 52.5.2. Let $(A, \mathfrak m)$ be a Noetherian local ring.

1. Let $(M_ n)$ be an inverse system of finite $A$-modules. Then the inverse system $H^ i_\mathfrak m(M_ n)$ satisfies the Mittag-Leffler condition for any $i$.

2. Let $U = \mathop{\mathrm{Spec}}(A) \setminus \{ \mathfrak m\}$ be the punctured spectrum of $A$. Let $\mathcal{F}_ n$ be an inverse system of coherent $\mathcal{O}_ U$-modules. Then the inverse system $H^ i(U, \mathcal{F}_ n)$ satisfies the Mittag-Leffler condition for $i > 0$.

Proof. Follows immediately from Lemma 52.5.1. $\square$

Lemma 52.5.3. Let $(A, \mathfrak m)$ be a Noetherian local ring. Let $(M_ n)$ be an inverse system of finite $A$-modules. Let $M \to \mathop{\mathrm{lim}}\nolimits M_ n$ be a map where $M$ is a finite $A$-module such that for some $i$ the map $H^ i_\mathfrak m(M) \to \mathop{\mathrm{lim}}\nolimits H^ i_\mathfrak m(M_ n)$ is an isomorphism. Then the inverse system $H^ i_\mathfrak m(M_ n)$ is essentially constant with value $H^ i_\mathfrak m(M)$.

Proof. By Lemma 52.5.2 the inverse system $H^ i_\mathfrak m(M_ n)$ satisfies the Mittag-Leffler condition. Let $E_ n \subset H^ i_\mathfrak m(M_ n)$ be the image of $H^ i_\mathfrak m(M_{n'})$ for $n' \gg n$. Then $(E_ n)$ is an inverse system with surjective transition maps and $H^ i_\mathfrak m(M) = \mathop{\mathrm{lim}}\nolimits E_ n$. Since $H^ i_\mathfrak m(M)$ has the descending chain condition by Lemma 52.5.1 we find there can only be a finite number of nontrivial kernels of the surjections $H^ i_\mathfrak m(M) \to E_ n$. Thus $E_ n \to E_{n - 1}$ is an isomorphism for all $n \gg 0$ as desired. $\square$

Lemma 52.5.4. Let $(A, \mathfrak m)$ be a Noetherian local ring. Let $I \subset A$ be an ideal. Let $M$ be a finite $A$-module. Then

$H^ i(R\Gamma _\mathfrak m(M)^\wedge ) = \mathop{\mathrm{lim}}\nolimits H^ i_\mathfrak m(M/I^ nM)$

for all $i$ where $R\Gamma _\mathfrak m(M)^\wedge$ denotes the derived $I$-adic completion.

Proof. Apply Dualizing Complexes, Lemma 47.12.4 and Lemma 52.5.2 to see the vanishing of the $R^1\mathop{\mathrm{lim}}\nolimits$ terms. $\square$

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