Lemma 20.39.1. 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 $E$ be an object of $D(\mathcal{O}_ X)$. Denote
\[ E_ n = E \otimes _{\mathcal{O}_ X} (\mathcal{O}_ X \xrightarrow {f^ n} \mathcal{O}_ X) \]
and set $E^\wedge = R\mathop{\mathrm{lim}}\nolimits E_ n$. For $p \in \mathbf{Z}$ is a canonical commutative diagram
\[ \xymatrix{ & 0 & 0 \\ 0 \ar[r] & \widehat{H^ p(X, E)} \ar[r] \ar[u] & \mathop{\mathrm{lim}}\nolimits H^ p(X, E_ n) \ar[r] \ar[u] & T_ f(H^{p + 1}(X, E)) \ar[r] & 0 \\ 0 \ar[r] & H^0(H^ p(X, E)^\wedge ) \ar[r] \ar[u] & H^ p(X, E^\wedge ) \ar[r] \ar[u] & T_ f(H^{p + 1}(X, E)) \ar[r] \ar@{=}[u] & 0 \\ & R^1\mathop{\mathrm{lim}}\nolimits H^ p(X, E)[f^ n] \ar[u] \ar[r]^\cong & R^1\mathop{\mathrm{lim}}\nolimits H^{p - 1}(X, E_ n) \ar[u] \\ & 0 \ar[u] & 0 \ar[u] } \]
with exact rows and columns where $\widehat{H^ p(X, E)} = \mathop{\mathrm{lim}}\nolimits H^ p(X, E)/f^ n H^ p(X, E)$ is the usual $f$-adic completion, $H^ p(X, E)^\wedge $ is the derived $f$-adic completion, and $T_ f(H^{p + 1}(X, E))$ is the $f$-adic Tate module, see More on Algebra, Example 15.93.5. Finally, we have $H^ p(X, E^\wedge ) = H^ p(R\Gamma (X, E)^\wedge )$.
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