The Stacks project

Example 15.90.22. Let $A$ be a ring and let $I \subset A$ be a finitely generated ideal. Let $K^\bullet $ be a complex of $A$-modules. We can apply Lemma 15.90.21 with $F^ pK^\bullet = \tau _{\leq -p}K^\bullet $. Then we get a bounded spectral sequence

\[ E_1^{p, q} = H^{p + q}(H^{-p}(K^\bullet )^\wedge [p]) = H^{2p + q}(H^{-p}(K^\bullet )^\wedge ) \]

converging to $H^{p + q}((K^\bullet )^\wedge )$. After renumbering $p = -j$ and $q = i + 2j$ we find that for any $K \in D(A)$ there is a bounded spectral sequence $(E'_ r, d'_ r)_{r \geq 2}$ of bigraded derived complete modules with $d'_ r$ of bidegree $(r, -r + 1)$, with

\[ (E'_2)^{i, j} = H^ i(H^ j(K)^\wedge ) \]

and converging to $H^{i + j}(K^\wedge )$.


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