The Stacks project

Lemma 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 $M$ be a finite $A$-module. Assume

  1. $A$ is $f$-adically complete,

  2. $H^1_\mathfrak a(M)$ and $H^2_\mathfrak a(M)$ are annihilated by a power of $f$.

Then with $U = \mathop{\mathrm{Spec}}(A) \setminus V(\mathfrak a)$ the map

\[ \Gamma (U, \widetilde{M}) \longrightarrow \mathop{\mathrm{lim}}\nolimits \Gamma (U, \widetilde{M/f^ nM}) \]

is an isomorphism.

Proof. We may apply Lemma 52.3.6 to $U$ and $\mathcal{F} = \widetilde{M}|_ U$ because $\mathcal{F}$ is a Noetherian object in the category of coherent $\mathcal{O}_ U$-modules. Since $H^1(U, \mathcal{F}) = H^2_\mathfrak a(M)$ (Local Cohomology, Lemma 51.8.2) is annihilated by a power of $f$, we see that its $f$-adic Tate module is zero. Hence the lemma shows $\mathop{\mathrm{lim}}\nolimits H^0(U, \mathcal{F}/f^ n \mathcal{F})$ is the $0$th cohomology group of the derived $f$-adic completion of $H^0(U, \mathcal{F})$. Consider the exact sequence

\[ 0 \to H^0_\mathfrak a(M) \to M \to \Gamma (U, \mathcal{F}) \to H^1_\mathfrak a(M) \to 0 \]

of Local Cohomology, Lemma 51.8.2. Since $H^1_\mathfrak a(M)$ is annihilated by a power of $f$ it is derived complete with respect to $(f)$. Since $M$ and $H^0_\mathfrak a(M)$ are finite $A$-modules they are complete (Algebra, Lemma 10.97.1) hence derived complete (More on Algebra, Proposition 15.91.5). By More on Algebra, Lemma 15.91.6 we conclude that $\Gamma (U, \mathcal{F})$ is derived complete as desired. $\square$


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