Example 15.91.5. Let $A$ be a ring and let $f \in A$. Denote $K \mapsto K^\wedge $ the derived completion with respect to $(f)$. Let $M$ be an $A$-module. Using that

by Lemma 15.90.18 and using Lemma 15.86.4 we obtain

the *$f$-adic Tate module of $M$*. Here the maps $M[f^ n] \to M[f^{n - 1}]$ are given by multiplication by $f$. Then there is a short exact sequence

describing $H^0(M^\wedge )$. We have $H^1(M^\wedge ) = R^1\mathop{\mathrm{lim}}\nolimits M/f^ nM = 0$ as the transition maps are surjective (Lemma 15.86.1). All the other cohomologies of $M^\wedge $ are zero for trivial reasons. We claim that for $K \in D(A)$ there are short exact sequences

Namely this follows from the spectral sequence of Example 15.90.22 because it degenerates at $E_2$ (as only $i = -1, 0$ give nonzero terms).

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