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. 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. $\square$

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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