Lemma 12.23.5. Let $\mathcal{A}$ be an abelian category. Let $(K, F, d)$ be a filtered differential object of $\mathcal{A}$. If $Z_\infty ^ p$ and $B_\infty ^ p$ exist (see proof), then

the limit $E_\infty $ exists and is graded having $E_\infty ^ p = Z_\infty ^ p/B_\infty ^ p$ in degree $p$, and

the associated graded $\text{gr}(H(K))$ of the cohomology of $K$ is a graded subquotient of the graded limit object $E_\infty $.

**Proof.**
The objects $Z_\infty $, $B_\infty $, and the limit $E_\infty = Z_\infty /B_\infty $ of Definition 12.20.2 are objects of $\text{Gr}(\mathcal{A})$ by our construction of the spectral sequence in the proof of Lemma 12.23.2. Since $Z_ r = \bigoplus Z_ r^ p$ and $B_ r = \bigoplus B_ r^ p$, if we assume that

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

and

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

exist, then $Z_\infty $ and $B_\infty $ exist with degree $p$ parts $Z_\infty ^ p$ and $B_\infty ^ p$ (follows from an elementary argument about unions and intersections of graded subobjects). Thus

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

where the top and bottom exist. We have

12.23.5.1
\begin{equation} \label{homology-equation-on-top} \mathop{\mathrm{Ker}}(d) \cap F^ pK + F^{p + 1}K \subset \bigcap \nolimits _ r \left(F^ pK \cap d^{-1}(F^{p + r}K) + F^{p + 1}K\right) \end{equation}

and

12.23.5.2
\begin{equation} \label{homology-equation-at-bottom} \bigcup \nolimits _ r \left(F^ pK \cap d(F^{p - r + 1}K) + F^{p + 1}K\right) \subset \mathop{\mathrm{Im}}(d) \cap F^ pK + F^{p + 1}K. \end{equation}

Thus a subquotient of $E_\infty ^ p$ is

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

Comparing with the formula given for $\text{gr}^ pH(K)$ in the discussion following Definition 12.23.4 we conclude.
$\square$

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