## 52.9 Algebraization of local cohomology, II

In this section we redo the arguments of Section 52.8 when $(A, \mathfrak m)$ is a local ring and we take local cohomology $R\Gamma _\mathfrak m$ with respect to $\mathfrak m$. As before our main tool is the following lemma.

Lemma 52.9.1. Let $(A, \mathfrak m)$ be a Noetherian local ring. Let $I \subset A$ be an ideal. Let $M$ be a finite $A$-module and let $\mathfrak p \subset A$ be a prime. Let $s$ and $d$ be integers. Assume

1. $A$ has a dualizing complex,

2. $\text{cd}(A, I) \leq d$, and

3. $\text{depth}_{A_\mathfrak p}(M_\mathfrak p) + \dim (A/\mathfrak p) > d + s$.

Then there exists an $f \in A \setminus \mathfrak p$ which annihilates $H^ i(R\Gamma _\mathfrak m(M)^\wedge )$ for $i \leq s$ where ${}^\wedge$ indicates $I$-adic completion.

Proof. According to Local Cohomology, Lemma 51.9.4 the function

$\mathfrak p' \longmapsto \text{depth}_{A_{\mathfrak p'}}(M_{\mathfrak p'}) + \dim (A/\mathfrak p')$

is lower semi-continuous on $\mathop{\mathrm{Spec}}(A)$. Thus the value of this function on $\mathfrak p' \subset \mathfrak p$ is $> s + d$. Thus our lemma is a special case of Lemma 52.8.1 provided that $\mathfrak p \not= \mathfrak m$. If $\mathfrak p = \mathfrak m$, then we have $H^ i_\mathfrak m(M) = 0$ for $i \leq s + d$ by the relationship between depth and local cohomology (Dualizing Complexes, Lemma 47.11.1). Thus the argument given in the proof of Lemma 52.8.1 shows that $H^ i(R\Gamma _\mathfrak m(M)^\wedge ) = 0$ for $i \leq s$ in this (degenerate) case. $\square$

Lemma 52.9.2. Let $(A, \mathfrak m)$ be a Noetherian local ring. Let $I \subset A$ be an ideal. Let $M$ be a finite $A$-module. Let $s$ and $d$ be integers. Assume

1. $A$ has a dualizing complex,

2. if $\mathfrak p \in V(I)$, then no condition,

3. if $\mathfrak p \not\in V(I)$ and $V(\mathfrak p) \cap V(I) = \{ \mathfrak m\}$, then $\dim (A/\mathfrak p) \leq d$,

4. if $\mathfrak p \not\in V(I)$ and $V(\mathfrak p) \cap V(I) \not= \{ \mathfrak m\}$, then

$\text{depth}_{A_\mathfrak p}(M_\mathfrak p) \geq s \quad \text{or}\quad \text{depth}_{A_\mathfrak p}(M_\mathfrak p) + \dim (A/\mathfrak p) > d + s$

Then there exists an ideal $J_0 \subset A$ with $V(J_0) \cap V(I) = \{ \mathfrak m\}$ such that for any $J \subset J_0$ with $V(J) \cap V(I) = \{ \mathfrak m\}$ the map

$R\Gamma _ J(M) \longrightarrow R\Gamma _{J_0}(M)$

induces an isomorphism in cohomology in degrees $\leq s$ and moreover these modules are annihilated by a power of $J_0I$.

Proof. This is a special case of Lemma 52.8.2. $\square$

Lemma 52.9.3. In Lemma 52.9.2 if instead of the empty condition (2) we assume

1. if $\mathfrak p \in V(I)$ and $\mathfrak p \not= \mathfrak m$, then $\text{depth}_{A_\mathfrak p}(M_\mathfrak p) + \dim (A/\mathfrak p) > s$,

then the conditions also imply that $H^ i_{J_0}(M)$ is a finite $A$-module for $i \leq s$.

Proof. This is a special case of Lemma 52.8.3. $\square$

Lemma 52.9.4. If in Lemma 52.9.2 we additionally assume

1. if $\mathfrak p \not\in V(I)$ and $V(\mathfrak p) \cap V(I) = \{ \mathfrak m\}$, then $\text{depth}_{A_\mathfrak p}(M_\mathfrak p) > s$,

then $H^ i_{J_0}(M) = H^ i_ J(M) = H^ i_\mathfrak m(M)$ for $i \leq s$ and these modules are annihilated by a power of $I$.

Proof. This is a special case of Lemma 52.8.4. $\square$

Lemma 52.9.5. Let $(A, \mathfrak m)$ be a Noetherian local ring. Let $I \subset A$ be an ideal. Let $M$ be a finite $A$-module. Let $s$ and $d$ be integers. Assume

1. $A$ is $I$-adically complete and has a dualizing complex,

2. if $\mathfrak p \in V(I)$, no condition,

3. $\text{cd}(A, I) \leq d$,

4. if $\mathfrak p \not\in V(I)$ and $V(\mathfrak p) \cap V(I) \not= \{ \mathfrak m\}$ then

$\text{depth}_{A_\mathfrak p}(M_\mathfrak p) \geq s \quad \text{or}\quad \text{depth}_{A_\mathfrak p}(M_\mathfrak p) + \dim (A/\mathfrak p) > d + s$

Then there exists an ideal $J_0 \subset A$ with $V(J_0) \cap V(I) = \{ \mathfrak m\}$ such that for any $J \subset J_0$ with $V(J) \cap V(I) = \{ \mathfrak m\}$ the map

$R\Gamma _ J(M) \longrightarrow R\Gamma _ J(M)^\wedge = R\Gamma _\mathfrak m(M)^\wedge$

induces an isomorphism in cohomology in degrees $\leq s$. Here ${}^\wedge$ denotes derived $I$-adic completion.

Proof. This lemma is a special case of Lemma 52.8.5 since condition (5)(c) is implied by condition (4) as $\delta _{max} = \delta _{min} = \delta (\mathfrak m)$. We will give the proof of this important special case as it is somewhat easier (fewer things to check).

There is no difference between $R\Gamma _\mathfrak a$ and $R\Gamma _{V(\mathfrak a)}$ in our current situation, see Dualizing Complexes, Lemma 47.10.1. Next, we observe that

$R\Gamma _\mathfrak m(M)^\wedge = R\Gamma _ I(R\Gamma _ J(M))^\wedge = R\Gamma _ J(M)^\wedge$

by Dualizing Complexes, Lemmas 47.9.6 and 47.12.1 which explains the equality sign in the statement of the lemma.

Observe that the lemma holds for $s < 0$. This is not a trivial case because it is not a priori clear that $H^ s(R\Gamma _\mathfrak m(M)^\wedge )$ is zero for negative $s$. However, this vanishing was established in Lemma 52.5.4. We will prove the lemma by induction for $s \geq 0$.

The assumptions of Lemma 52.9.2 are satisfied by Local Cohomology, Lemma 51.4.10. The lemma for $s = 0$ follows from Lemma 52.9.2 and Dualizing Complexes, Lemma 47.12.5.

Assume $s > 0$ and the lemma holds for smaller values of $s$. Let $M' \subset M$ be the submodule of elements whose support is condained in $V(I) \cup V(J)$ for some ideal $J$ with $V(J) \cap V(I) = \{ \mathfrak m\}$. Then $M'$ is a finite $A$-module. We claim that

$R\Gamma _ J(M') \to R\Gamma _\mathfrak m(M')^\wedge$

is an isomorphism for any choice of $J$. Namely, for any such module there is a short exact sequence $0 \to M_1 \oplus M_2 \to M' \to N \to 0$ with $M_1$ annihilated by a power of $J$, with $M_2$ annihilated by a power of $I$ and with $N$ annihilated by a power of $\mathfrak m$. In the case of $M_1$ we see that $R\Gamma _ J(M_1) = M_1$ and since $M_1$ is a finite $A$-module and $I$-adically complete we have $M_1^\wedge = M_1$. Thus the claim holds for $M_1$. In the case of $M_2$ we see that $H^ i_ J(M_2)$ is annihilated by a power of $I$ and hence derived complete. Thus the claim for $M_2$. By the same arguments the claim holds for $N$ and we conclude that the claim holds. Considering the short exact sequence $0 \to M' \to M \to M/M' \to 0$ we see that it suffices to prove the lemma for $M/M'$. This we may assume $\mathfrak p \in \text{Ass}(M)$ implies $V(\mathfrak p) \cap V(I) \not= \{ \mathfrak m\}$, i.e., $\mathfrak p$ is a prime as in (4).

Choose an ideal $J_0$ as in Lemma 52.9.2 and an integer $t > 0$ such that $(J_0I)^ t$ annihilates $H^ s_ J(M)$. Here $J$ denotes an arbitrary ideal $J \subset J_0$ with $V(J) \cap V(I) = \{ \mathfrak m\}$. The assumptions of Lemma 52.9.1 are satisfied for every $\mathfrak p \in \text{Ass}(M)$ (see previous paragraph). Thus the annihilator $\mathfrak a \subset A$ of $H^ s(R\Gamma _\mathfrak m(M)^\wedge )$ is not contained in $\mathfrak p$ for $\mathfrak p \in \text{Ass}(M)$. Thus we can find an $f \in \mathfrak a(J_0I)^ t$ not in any associated prime of $M$ which is an annihilator of both $H^ s(R\Gamma _\mathfrak m(M)^\wedge )$ and $H^ s_ J(M)$. Then $f$ is a nonzerodivisor on $M$ and we can consider the short exact sequence

$0 \to M \xrightarrow {f} M \to M/fM \to 0$

Our choice of $f$ shows that we obtain

$\xymatrix{ H^{s - 1}_ J(M) \ar[d] \ar[r] & H^{s - 1}_ J(M/fM) \ar[d] \ar[r] & H^ s_ J(M) \ar[d] \ar[r] & 0 \\ H^{s - 1}(R\Gamma _\mathfrak m(M)^\wedge ) \ar[r] & H^{s - 1}(R\Gamma _\mathfrak m(M/fM)^\wedge ) \ar[r] & H^ s(R\Gamma _\mathfrak m(M)^\wedge ) \ar[r] & 0 }$

for any $J \subset J_0$ with $V(J) \cap V(I) = \{ \mathfrak m\}$. Thus if we choose $J$ such that it works for $M$ and $M/fM$ and $s - 1$ (possible by induction hypothesis), then we conclude that the lemma is true. $\square$

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