Example 15.91.22 (0BKE)—The Stacks project

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Table of contents
Part 1: Preliminaries
Chapter 15: More on Algebra
Section 15.91: Derived Completion
Example 15.91.22 (cite)

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Example 15.91.22. 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 15.91.21 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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