The Stacks project

Proposition 52.12.6. Let $A$ be a Noetherian ring. Let $f \in \mathfrak a \subset A$ be an element of an ideal of $A$. Let $\mathcal{F}$ be a coherent module on $U = \mathop{\mathrm{Spec}}(A) \setminus V(\mathfrak a)$. Assume

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

  2. if $x \in \text{Ass}(\mathcal{F})$, $x \not\in V(f)$, $\overline{\{ x\} } \cap V(f) \not\subset V(\mathfrak a)$, and $z \in \overline{\{ x\} } \cap V(\mathfrak a)$, then $\dim (\mathcal{O}_{\overline{\{ x\} }, z}) > 2$.

Then the map

\[ \mathop{\mathrm{colim}}\nolimits _ V \Gamma (V, \mathcal{F}) \longrightarrow \mathop{\mathrm{lim}}\nolimits \Gamma (U, \mathcal{F}/f^ n\mathcal{F}) \]

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

First proof. Recall that $A$ is universally catenary and with Gorenstein formal fibres, see Dualizing Complexes, Lemmas 47.23.2 and 47.17.4. Thus we may consider the map $\mathcal{F} \to \mathcal{F}'$ constructed in Local Cohomology, Lemma 51.15.3 for the closed subset $V(f) \cap U$ of $U$. Observe that

  1. The kernel and cokernel of $\mathcal{F} \to \mathcal{F}'$ are supported on $V(f) \cap U$.

  2. The module $\mathcal{F}'$ is $f$-torsion free as its stalks have depth $\geq 1$ for all points of $V(f) \cap U$, i.e., $\mathcal{F}'$ has no associated points in $V(f) \cap U$.

  3. If $y \in V(f) \cap U$ is an associated point of $\mathcal{F}'/f\mathcal{F}'$, then $\text{depth}(\mathcal{F}'_ y) = 1$ and hence (by the construction of $\mathcal{F}'$) there is an immediate specialization $x \leadsto y$ with $x \not\in V(f)$ an associated point of $\mathcal{F}$. It follows that $y$ cannot have an immediate specialization in $\mathop{\mathrm{Spec}}(A)$ to a point $z \in V(\mathfrak a)$ by our assumption (2).

  4. It follows from (3) that $H^0(U, \mathcal{F}'/f\mathcal{F}')$ is a finite $A$-module, see Local Cohomology, Lemma 51.12.1.

These observations will allow us to finish the proof.

First, we claim the lemma holds for $\mathcal{F}'$. Namely, 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. Since $\mathcal{F}'$ is $f$-torsion free, we may assume $M'$ is $f$-torsion free as well. Observation (4) above shows that $H^1_\mathfrak a(M')$ is a finite $A$-module, see Local Cohomology, Lemma 51.8.2. Thus the claim by Lemma 52.12.5.

Second, we observe that the lemma holds trivially for any coherent $\mathcal{O}_ U$-module supported on $V(f) \cap U$. Let $\mathcal{K}$, resp. $\mathcal{G}$, resp. $\mathcal{Q}$ be the kernel, resp. image, resp. cokernel of the map $\mathcal{F} \to \mathcal{F}'$. The short exact sequence $0 \to \mathcal{G} \to \mathcal{F}' \to \mathcal{Q} \to 0$ and Lemma 52.12.1 show that the result holds for $\mathcal{G}$. Then we do this again with the short exact sequence $0 \to \mathcal{K} \to \mathcal{F} \to \mathcal{G} \to 0$ to finish the proof. $\square$

Second proof. The proposition is a special case of Proposition 52.12.3. $\square$

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