Lemma 52.12.1. Let $I \subset \mathfrak a$ be ideals of a Noetherian ring $A$. Let $\mathcal{F}$ be a coherent module on $U = \mathop{\mathrm{Spec}}(A) \setminus V(\mathfrak a)$. Assume

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

if $x \in \text{Ass}(\mathcal{F})$, $x \not\in V(I)$, $\overline{\{ x\} } \cap V(I) \not\subset V(\mathfrak a)$ and $z \in \overline{\{ x\} } \cap V(\mathfrak a)$, then $\dim (\mathcal{O}_{\overline{\{ x\} }, z}) > \text{cd}(A, I) + 1$,

one of the following holds:

the restriction of $\mathcal{F}$ to $U \setminus V(I)$ is $(S_1)$

the dimension of $V(\mathfrak a)$ is at most $2$^{1}.

Then we obtain an isomorphism

\[ \mathop{\mathrm{colim}}\nolimits H^0(V, \mathcal{F}) \longrightarrow \mathop{\mathrm{lim}}\nolimits H^0(U, \mathcal{F}/I^ n\mathcal{F}) \]

where the colimit is over opens $V \subset U$ containing $U \cap V(I)$.

**Proof.**
Choose a finite $A$-module $M$ such that $\mathcal{F}$ is the restriction to $U$ of the coherent module associated to $M$, see Local Cohomology, Lemma 51.8.2. Set $d = \text{cd}(A, I)$. Let $\mathfrak p$ be a prime of $A$ not contained in $V(I)$ and let $\mathfrak q \in V(\mathfrak p) \cap V(\mathfrak a)$. Then either $\mathfrak p$ is not an associated prime of $M$ and hence $\text{depth}(M_\mathfrak p) \geq 1$ or we have $\dim ((A/\mathfrak p)_\mathfrak q) > d + 1$ by (2). Thus the hypotheses of Lemma 52.8.5 are satisfied for $s = 1$ and $d$; here we use condition (3). Thus we find there exists an ideal $J_0 \subset \mathfrak a$ with $V(J_0) \cap V(I) = V(\mathfrak a)$ such that for any $J \subset J_0$ with $V(J) \cap V(I) = V(\mathfrak a)$ the maps

\[ H^ i_ J(M) \longrightarrow H^ i(R\Gamma _\mathfrak a(M)^\wedge ) \]

are isomorphisms for $i = 0, 1$. Consider the morphisms of exact triangles

\[ \xymatrix{ R\Gamma _ J(M) \ar[d] \ar[r] & M \ar[r] \ar[d] & R\Gamma (V, \mathcal{F}) \ar[d] \ar[r] & R\Gamma _ J(M)[1] \ar[d] \\ R\Gamma _ J(M)^\wedge \ar[r] & M \ar[r] & R\Gamma (V, \mathcal{F})^\wedge \ar[r] & R\Gamma _ J(M)^\wedge [1] \\ R\Gamma _\mathfrak a(M)^\wedge \ar[r] \ar[u] & M \ar[r] \ar[u] & R\Gamma (U, \mathcal{F})^\wedge \ar[r] \ar[u] & R\Gamma _\mathfrak a(M)^\wedge [1] \ar[u] } \]

where $V = \mathop{\mathrm{Spec}}(A) \setminus V(J)$. Recall that $R\Gamma _\mathfrak a(M)^\wedge \to R\Gamma _ J(M)^\wedge $ is an isomorphism (because $\mathfrak a$, $\mathfrak a + I$, and $J + I$ cut out the same closed subscheme, for example see proof of Lemma 52.8.5). Hence $R\Gamma (U, \mathcal{F})^\wedge = R\Gamma (V, \mathcal{F})^\wedge $. This produces a commutative diagram

\[ \xymatrix{ 0 \ar[r] & H^0_ J(M) \ar[r] \ar[d] & M \ar[r] \ar[d] \ar[r] & \Gamma (V, \mathcal{F}) \ar[d] \ar[r] & H^1_ J(M) \ar[d] \ar[r] & 0 \\ 0 \ar[r] & H^0(R\Gamma _ J(M)^\wedge ) \ar[r] & M \ar[r] & H^0(R\Gamma (V, \mathcal{F})^\wedge ) \ar[r] & H^1(R\Gamma _ J(M)^\wedge ) \ar[r] & 0 \\ 0 \ar[r] & H^0(R\Gamma _\mathfrak a(M)^\wedge ) \ar[r] \ar[u] & M \ar[r] \ar[u] & H^0(R\Gamma (U, \mathcal{F})^\wedge ) \ar[r] \ar[u] & H^1(R\Gamma _\mathfrak a(M)^\wedge ) \ar[r] \ar[u] & 0 } \]

with exact rows and isomorphisms for the lower vertical arrows. Hence we obtain an isomorphism $\Gamma (V, \mathcal{F}) \to H^0(R\Gamma (U, \mathcal{F})^\wedge )$. By Lemmas 52.6.20 and 52.7.2 we have

\[ R\Gamma (U, \mathcal{F})^\wedge = R\Gamma (U, \mathcal{F}^\wedge ) = R\Gamma (U, R\mathop{\mathrm{lim}}\nolimits \mathcal{F}/I^ n\mathcal{F}) \]

and we find $H^0(R\Gamma (U, \mathcal{F})^\wedge ) = \mathop{\mathrm{lim}}\nolimits H^0(U, \mathcal{F}/I^ n\mathcal{F})$ by Cohomology, Lemma 20.36.1.
$\square$

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