\begin{equation*} \DeclareMathOperator\Coim{Coim} \DeclareMathOperator\Coker{Coker} \DeclareMathOperator\Ext{Ext} \DeclareMathOperator\Hom{Hom} \DeclareMathOperator\Im{Im} \DeclareMathOperator\Ker{Ker} \DeclareMathOperator\Mor{Mor} \DeclareMathOperator\Ob{Ob} \DeclareMathOperator\Sh{Sh} \DeclareMathOperator\SheafExt{\mathcal{E}\mathit{xt}} \DeclareMathOperator\SheafHom{\mathcal{H}\mathit{om}} \DeclareMathOperator\Spec{Spec} \newcommand\colim{\mathop{\mathrm{colim}}\nolimits} \newcommand\lim{\mathop{\mathrm{lim}}\nolimits} \newcommand\Qcoh{\mathit{Qcoh}} \newcommand\Sch{\mathit{Sch}} \newcommand\QCohstack{\mathcal{QC}\!\mathit{oh}} \newcommand\Cohstack{\mathcal{C}\!\mathit{oh}} \newcommand\Spacesstack{\mathcal{S}\!\mathit{paces}} \newcommand\Quotfunctor{\mathrm{Quot}} \newcommand\Hilbfunctor{\mathrm{Hilb}} \newcommand\Curvesstack{\mathcal{C}\!\mathit{urves}} \newcommand\Polarizedstack{\mathcal{P}\!\mathit{olarized}} \newcommand\Complexesstack{\mathcal{C}\!\mathit{omplexes}} \newcommand\Pic{\mathop{\mathrm{Pic}}\nolimits} \newcommand\Picardstack{\mathcal{P}\!\mathit{ic}} \newcommand\Picardfunctor{\mathrm{Pic}} \newcommand\Deformationcategory{\mathcal{D}\!\mathit{ef}} \end{equation*}

The Stacks project

[Lemma 1.6, Bhatt-local]

Lemma 49.3.6. Let $A$ be a ring and $f \in A$. Let $X$ be a scheme over $A$. Let $\mathcal{F}$ be a quasi-coherent $\mathcal{O}_ X$-module. Assume that $\mathcal{F}[f^ n] = \mathop{\mathrm{Ker}}(f^ n : \mathcal{F} \to \mathcal{F})$ stabilizes. Then

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

where the right hand side indicates the derived completion with respect to the ideal $(f) \subset A$. Let $H^ p$ be the $p$th cohomology group of this complex. Then there are short exact sequences

\[ 0 \to R^1\mathop{\mathrm{lim}}\nolimits H^{p - 1}(X, \mathcal{F}/f^ n\mathcal{F}) \to H^ p \to \mathop{\mathrm{lim}}\nolimits H^ p(X, \mathcal{F}/f^ n\mathcal{F}) \to 0 \]

and

\[ 0 \to H^0(H^ p(X, \mathcal{F})^\wedge ) \to H^ p \to T_ f(H^{p + 1}(X, \mathcal{F})) \to 0 \]

where $T_ f(-)$ denote the $f$-adic Tate module as in More on Algebra, Example 15.83.4.

Proof. We start with the canonical identifications

\begin{align*} R\Gamma (X, \mathcal{F})^\wedge & = R\mathop{\mathrm{lim}}\nolimits R\Gamma (X, \mathcal{F}) \otimes _ A^\mathbf {L} (A \xrightarrow {f^ n} A) \\ & = R\mathop{\mathrm{lim}}\nolimits R\Gamma (X, \mathcal{F} \xrightarrow {f^ n} \mathcal{F}) \\ & = R\Gamma (X, R\mathop{\mathrm{lim}}\nolimits (\mathcal{F} \xrightarrow {f^ n} \mathcal{F})) \end{align*}

The first equality holds by More on Algebra, Lemma 15.82.17. The second by the projection formula, see Cohomology, Lemma 20.45.3. The third by Cohomology, Lemma 20.32.2. Note that by Derived Categories of Schemes, Lemma 35.3.2 we have $\mathop{\mathrm{lim}}\nolimits \mathcal{F}/f^ n\mathcal{F} = R\mathop{\mathrm{lim}}\nolimits \mathcal{F}/f^ n \mathcal{F}$. Thus to finish the proof of the first statement of the lemma it suffices to show that the pro-objects $(f^ n : \mathcal{F} \to \mathcal{F})$ and $(\mathcal{F}/f^ n \mathcal{F})$ are isomorphic. There is clearly a map from the first inverse system to the second. Suppose that $\mathcal{F}[f^ c] = \mathcal{F}[f^{c + 1}] = \mathcal{F}[f^{c + 2}] = \ldots $. Then we can define an arrow of inverse systems in $D(\mathcal{O}_ X)$ in the other direction by the diagrams

\[ \xymatrix{ \mathcal{F}/\mathcal{F}[f^ c] \ar[r]_-{f^{n + c}} \ar[d]_{f^ c} & \mathcal{F} \ar[d]^1 \\ \mathcal{F} \ar[r]^{f^ n} & \mathcal{F} } \]

Since the top horizontal arrow is injective the complex in the top row is quasi-isomorphic to $\mathcal{F}/f^{n + c}\mathcal{F}$. Some details omitted.

Since $R\Gamma (X, -)$ commutes with derived limits (Injectives, Lemma 19.13.6) we see that

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

(for first equality see first paragraph of proof). By More on Algebra, Remark 15.77.9 we obtain exact sequences

\[ 0 \to R^1\mathop{\mathrm{lim}}\nolimits H^{p - 1}(X, \mathcal{F}/f^ n\mathcal{F}) \to H^ p(X, \mathop{\mathrm{lim}}\nolimits \mathcal{F}/I^ n\mathcal{F}) \to \mathop{\mathrm{lim}}\nolimits H^ p(X, \mathcal{F}/I^ n\mathcal{F}) \to 0 \]

of $A$-modules. The second set of short exact sequences follow immediately from the discussion in More on Algebra, Example 15.83.4. $\square$


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