Lemma 52.8.1. Let $I, J$ be ideals of a Noetherian ring $A$. Let $M$ be a finite $A$-module. Let $\mathfrak p \subset A$ be a prime. Let $s$ and $d$ be integers. Assume
$A$ has a dualizing complex,
$\mathfrak p \not\in V(J) \cap V(I)$,
$\text{cd}(A, I) \leq d$, and
for all primes $\mathfrak p' \subset \mathfrak p$ we have $\text{depth}_{A_{\mathfrak p'}}(M_{\mathfrak p'}) + \dim ((A/\mathfrak p')_\mathfrak q) > d + s$ for all $\mathfrak q \in V(\mathfrak p') \cap V(J) \cap V(I)$.
Then there exists an $f \in A$, $f \not\in \mathfrak p$ which annihilates $H^ i(R\Gamma _ J(M)^\wedge )$ for $i \leq s$ where ${}^\wedge $ indicates $I$-adic completion.
Proof.
We will use that $R\Gamma _ J = R\Gamma _{V(J)}$ and similarly for $I + J$, see Dualizing Complexes, Lemma 47.10.1. Observe that $R\Gamma _ J(M)^\wedge = R\Gamma _ I(R\Gamma _ J(M))^\wedge = R\Gamma _{I + J}(M)^\wedge $, see Dualizing Complexes, Lemmas 47.12.1 and 47.9.6. Thus we may replace $J$ by $I + J$ and assume $I \subset J$ and $\mathfrak p \not\in V(J)$. Recall that
\[ R\Gamma _ J(M)^\wedge = R\mathop{\mathrm{Hom}}\nolimits _ A(R\Gamma _ I(A), R\Gamma _ J(M)) \]
by the description of derived completion in More on Algebra, Lemma 15.91.10 combined with the description of local cohomology in Dualizing Complexes, Lemma 47.10.2. Assumption (3) means that $R\Gamma _ I(A)$ has nonzero cohomology only in degrees $\leq d$. Using the canonical truncations of $R\Gamma _ I(A)$ we find it suffices to show that
\[ \text{Ext}^ i(N, R\Gamma _ J(M)) \]
is annihilated by an $f \in A$, $f \not\in \mathfrak p$ for $i \leq s + d$ and any $A$-module $N$. In turn using the canonical truncations for $R\Gamma _ J(M)$ we see that it suffices to show $H^ i_ J(M)$ is annihilated by an $f \in A$, $f \not\in \mathfrak p$ for $i \leq s + d$. This follows from Local Cohomology, Lemma 51.10.2.
$\square$
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