Lemma 12.23.5 (012F)—The Stacks project

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. $\square$

Comments (4)

Comment #1452 by Zhe Zhang on May 11, 2015 at 06:45

I can't see why $\text{gr}^p H(K) = \frac{\Ker(d) \cap F^pK + F^{p + 1}K} {\Im(d) \cap F^pK + F^{p + 1}K}$ Since by definition (if I'm not mistaken) this stuff should be $F^pH(K)/F^{p+1}H(K)$ where $F^pH(K)= \frac{\Ker(d) \cap F^pK + \Im(d)} {\Im(d) \cap F^pK + \Im(d)}$ Hence the quotient should be $\frac{\Ker(d) \cap F^pK + \Im(d)} {\Ker(d) \cap F^{p+1}K + \Im(d)}$ But then the inclusion in (12.20.5.1) (and the next) cannot hold in general, unless one put some extra conditions here, for example, the filtration on K is exhaustive.

Comment #1454 by Zhe Zhang on May 11, 2015 at 08:05

...and the inclusions are incorrect.(12.20.5.1) should be $\Ker(d) \cap F^pK + F^{p + 1}K \subset \bigcap\nolimits_r \left(F^pK \cap d(F^{p - r + 1}K) + F^{p + 1}K\right)$ and (12.20.5.2) should be $\bigcup\nolimits_r \left(F^pK \cap d^{-1}(F^{p + r}K) + F^{p + 1}K\right) \subset \Im(d) \cap F^pK + F^{p + 1}K.$

Comment #1455 by Zhe Zhang on May 11, 2015 at 08:31

A typo in the Comment #1452 $F^pH(K)= \frac{\Ker(d) \cap F^pK + \Im(d)} {\Im(d)}$

Comment #1462 by Johan on May 15, 2015 at 15:27

Thanks very much. It turns out the lemma is correct anyway without extra assumptions (because somehow the two mistakes you pointed out cancelled each other). The fix is here. The project will be updated later today and then you'll be able to read it online.

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

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