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

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

and

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

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

12.24.6.1
$$\label{homology-equation-on-top-bigraded} \mathop{\mathrm{Ker}}(d) \cap F^ pK^{n} + F^{p + 1}K^{n} \subset \bigcap \nolimits _ r \left( F^ pK^{n} \cap d^{-1}(F^{p + r}K^{n + 1}) + F^{p + 1}K^{n} \right)$$

and

12.24.6.2
$$\label{homology-equation-at-bottom-bigraded} \bigcup \nolimits _ r \left( F^ pK^{n} \cap d(F^{p - r + 1}K^{n - 1}) + F^{p + 1}K^{n} \right) \subset \mathop{\mathrm{Im}}(d) \cap F^ pK^{n} + F^{p + 1}K^{n}.$$

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 (12.24.5.2) we conclude. $\square$

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