The Stacks project

Example 15.93.5. 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 15.91.18 and using Lemma 15.87.4 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 15.87.1). All the other cohomologies of $M^\wedge $ are zero for trivial reasons. Finally, for $K \in D(A)$ and $p \in \mathbf{Z}$ there is a short exact sequence

This follows from the spectral sequence of Example 15.91.22 because it degenerates at $E_2$ (as only $i = -1, 0$ give nonzero terms); the next lemma gives more information.