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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

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

  2. 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 (5)

Comment #1452 by Zhe Zhang on

I can't see why Since by definition (if I'm not mistaken) this stuff should be where Hence the quotient should be 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

...and the inclusions are incorrect.(12.20.5.1) should be and (12.20.5.2) should be

Comment #1455 by Zhe Zhang on

A typo in the Comment #1452

Comment #1462 by on

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.

Comment #9487 by on

  1. In the statement, instead of “the limit exists” shouldn't we specify “the limit of the spectral sequence of Lemma 12.23.2 exists”?
  2. In the proof, second sentence, instead of , , shouldn't one write , ? (We are working over an abelian category that might not satisfy AB3.)

There are also:

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

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