The Stacks project

Lemma 52.12.5. 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. $f$ is a nonzerodivisor on $M$,

  3. $H^1_\mathfrak a(M/fM)$ is a finite $A$-module.

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

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

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

Proof. Set $\mathcal{F} = \widetilde{M}|_ U$. The finiteness of $H^1_\mathfrak a(M/fM)$ implies that $H^0(U, \mathcal{F}/f\mathcal{F})$ is finite, see Local Cohomology, Lemma 51.8.2. By Cohomology, Lemma 20.36.3 (which applies as $f$ is a nonzerodivisor on $\mathcal{F}$) we see that $N = \mathop{\mathrm{lim}}\nolimits H^0(U, \mathcal{F}/f^ n\mathcal{F})$ is a finite $A$-module, is $f$-torsion free, and $N/fN \subset H^0(U, \mathcal{F}/f\mathcal{F})$. On the other hand, we have a map $M \to N$ and a compatible map

\[ M/fM \longrightarrow H^0(U, \mathcal{F}/f\mathcal{F}) \]

For $g \in \mathfrak a$ we see that $(M/fM)_ g$ maps isomorphically to $H^0(U \cap D(f), \mathcal{F}/f\mathcal{F})$ since $\mathcal{F}/f\mathcal{F}$ is the restriction of $\widetilde{M/fM}$ to $U$. We conclude that $M_ g \to N_ g$ induces an isomorphism

\[ M_ g/fM_ g = (M/fM)_ g \to (N/fN)_ g = N_ g/fN_ g \]

Since $f$ is a nonzerodivisor on both $N$ and $M$ we conclude that $M_ g \to N_ g$ induces an isomorphism on $f$-adic completions which in turn implies $M_ g \to N_ g$ is an isomorphism in an open neightbourhood of $V(f) \cap D(g)$. Since $g \in \mathfrak a$ was arbitrary, we conclude that $M$ and $N$ determine isomorphic coherent modules over an open $V$ as in the statement of the lemma. This finishes the proof. $\square$

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