The Stacks project

Lemma 12.24.6. Let $\mathcal{A}$ be an abelian category. Let $(K^\bullet , F)$ be a filtered complex of $\mathcal{A}$. If $Z_\infty ^{p, q}$ and $B_\infty ^{p, q}$ exist (see proof), then

  1. the limit $E_\infty $ exists and is a bigraded object having $E_\infty ^{p, q} = Z_\infty ^{p, q}/B_\infty ^{p, q}$ in bidegree $(p, q)$,

  2. the $p$th graded part $\text{gr}^ pH^ n(K^\bullet )$ of the $n$th cohomology object of $K^\bullet $ is a subquotient of $E_\infty ^{p, n - p}$.

Proof. The objects $Z_\infty $, $B_\infty $, and the limit $E_\infty = Z_\infty /B_\infty $ of Definition 12.20.2 are bigraded objects of $\mathcal{A}$ by our construction of the spectral sequence in Lemma 12.24.2. Since $Z_ r = \bigoplus Z_ r^{p, q}$ and $B_ r = \bigoplus B_ r^{p, q}$, if we assume that

\[ Z_\infty ^{p, q} = \bigcap \nolimits _ r Z_ r^{p, q} = \bigcap \nolimits _ r \frac{F^ pK^{p + q} \cap d^{-1}(F^{p + r}K^{p + q + 1}) + F^{p + 1}K^{p + q}}{F^{p + 1}K^{p + q}} \]


\[ B_\infty ^{p, q} = \bigcup \nolimits _ r B_ r^{p, q} = \bigcup \nolimits _ r \frac{F^ pK^{p + q} \cap d(F^{p - r + 1}K^{p + q - 1}) + F^{p + 1}K^{p + q}}{F^{p + 1}K^{p + q}} \]

exist, then $Z_\infty $ and $B_\infty $ exist with bidegree $(p, q)$ parts $Z_\infty ^{p, q}$ and $B_\infty ^{p, q}$ (follows from an elementary argument about unions and intersections of bigraded objects). Thus

\[ E_\infty ^{p, q} = \frac{\bigcap _ r (F^ pK^{p + q} \cap d^{-1}(F^{p + r}K^{p + q + 1}) + F^{p + 1}K^{p + q})}{\bigcup _ r (F^ pK^{p + q} \cap d(F^{p - r + 1}K^{p + q - 1}) + F^{p + 1}K^{p + q})}. \]

where the top and the bottom exist. With $n = p + q$ we have
\begin{equation} \label{homology-equation-on-top-bigraded} \mathop{\mathrm{Ker}}(d) \cap F^ pK^{n} + F^{p + 1}K^{n} \subset \bigcap \nolimits _ r \left( F^ pK^{n} \cap d^{-1}(F^{p + r}K^{n + 1}) + F^{p + 1}K^{n} \right) \end{equation}

\begin{equation} \label{homology-equation-at-bottom-bigraded} \bigcup \nolimits _ r \left( F^ pK^{n} \cap d(F^{p - r + 1}K^{n - 1}) + F^{p + 1}K^{n} \right) \subset \mathop{\mathrm{Im}}(d) \cap F^ pK^{n} + F^{p + 1}K^{n}. \end{equation}

Thus a subquotient of $E_\infty ^{p, q}$ is

\[ \frac{\mathop{\mathrm{Ker}}(d) \cap F^ pK^{n} + F^{p + 1}K^ n}{\mathop{\mathrm{Im}}(d) \cap F^ pK^{n} + F^{p + 1}K^{n}} = \frac{\mathop{\mathrm{Ker}}(d) \cap F^ pK^ n}{\mathop{\mathrm{Im}}(d) \cap F^ pK^ n + \mathop{\mathrm{Ker}}(d) \cap F^{p + 1}K^ n} \]

Comparing with ( we conclude. $\square$

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