Example 15.83.4. 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 15.82.17 and using Lemma 15.78.4 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 15.78.1). 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 15.82.20 because it degenerates at $E_2$ (as only $i = -1, 0$ give nonzero terms).
Comments (0)