Example 20.39.3. Let $(X, \mathcal{O}_ X)$ be a ringed space. Let $A \to \Gamma (X, \mathcal{O}_ X)$ be a ring map and let $f \in A$. Let $\mathcal{F}$ be an $\mathcal{O}_ X$-module. Assume there is a $c$ such that $\mathcal{F}[f^ c] = \mathcal{F}[f^ n]$ for all $n \geq c$. We are going to apply Lemma 20.39.1 with $E = \mathcal{F}$. By Lemma 20.39.2 we see that the inverse system $(E_ n)$ is pro-isomorphic to the inverse system $(\mathcal{F}/f^ n\mathcal{F})$. We conclude that for $p \in \mathbf{Z}$ we obtain a commutative diagram

$\xymatrix{ & 0 & 0 \\ 0 \ar[r] & \widehat{H^ p(X, \mathcal{F})} \ar[r] \ar[u] & \mathop{\mathrm{lim}}\nolimits H^ p(X, \mathcal{F}/f^ n\mathcal{F}) \ar[r] \ar[u] & T_ f(H^{p + 1}(X, \mathcal{F})) \ar[r] & 0 \\ 0 \ar[r] & H^0(H^ p(X, \mathcal{F})^\wedge ) \ar[r] \ar[u] & H^ p(R\Gamma (X, \mathcal{F})^\wedge ) \ar[r] \ar[u] & T_ f(H^{p + 1}(X, \mathcal{F})) \ar[r] \ar@{=}[u] & 0 \\ & R^1\mathop{\mathrm{lim}}\nolimits H^ p(X, \mathcal{F})[f^ n] \ar[u] \ar[r]^\cong & R^1\mathop{\mathrm{lim}}\nolimits H^{p - 1}(X, \mathcal{F}/f^ n\mathcal{F}) \ar[u] \\ & 0 \ar[u] & 0 \ar[u] }$

with exact rows and columns where $\widehat{H^ p(X, \mathcal{F})} = \mathop{\mathrm{lim}}\nolimits H^ p(X, \mathcal{F})/f^ n H^ p(X, \mathcal{F})$ is the usual $f$-adic completion and $M^\wedge$ denotes derived $f$-adic completion for $M$ in $D(A)$.

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