The Stacks project

Lemma 12.23.3. Let $\mathcal{A}$ be an abelian category. Let $(K, F, d)$ be a filtered differential object of $\mathcal{A}$. The spectral sequence $(E_ r, d_ r)_{r \geq 0}$ associated to $(K, F, d)$ has

\[ d_1^ p : E_1^ p = H(\text{gr}^ pK) \longrightarrow H(\text{gr}^{p + 1}K) = E_1^{p + 1} \]

equal to the boundary map in homology associated to the short exact sequence of differential objects

\[ 0 \to \text{gr}^{p + 1}K \to F^ pK/F^{p + 2}K \to \text{gr}^ pK \to 0. \]

Proof. This is clear from the formula for the differential $d_1^ p$ given just above Lemma 12.23.2. $\square$

Comments (2)

Comment #1453 by Zhe Zhang on

The short exact sequence is incorrect. The last term should be

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  • 5 comment(s) on Section 12.23: Spectral sequences: filtered differential objects

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