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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