Definition 12.23.1. Let $\mathcal{A}$ be an abelian category. A filtered differential object $(K, F, d)$ is a filtered object $(K, F)$ of $\mathcal{A}$ endowed with an endomorphism $d : (K, F) \to (K, F)$ whose square is zero: $d \circ d = 0$.
12.23 Spectral sequences: filtered differential objects
We can build a spectral sequence starting with a filtered differential object.
To describe the spectral sequence associated to such an object we assume, for the moment, that $\mathcal{A}$ is an abelian category which has countable direct sums and countable direct sums are exact (this is not automatic, see Remark 12.16.3). Let $(K, F, d)$ be a filtered differential object of $\mathcal{A}$. Note that each $F^ nK$ is a differential object by itself. Consider the object $A = \bigoplus F^ nK$ and endow it with a differential $d$ by using $d$ on each summand. Then $(A, d)$ is a differential object of $\mathcal{A}$ which comes equipped with a grading. Consider the map
\[ \alpha : A \to A \]
which is given by the inclusions $F^ nK \to F^{n - 1}K$. This is clearly an injective morphism of differential objects $\alpha : (A, d) \to (A, d)$. Hence, by Definition 12.22.5 we get a spectral sequence. We will call this the spectral sequence associated to the filtered differential object $(K, F, d)$.
Let us figure out the terms of this spectral sequence. First, note that $A/\alpha A = \text{gr}(K)$ endowed with its differential $d = \text{gr}(d)$. Hence we see that
\[ E_0 = \text{gr}(K), \quad d_0 = \text{gr}(d). \]
Hence the homology of the graded differential object $\text{gr}(K)$ is the next term:
\[ E_1 = H(\text{gr}(K), \text{gr}(d)). \]
In addition we see that $E_0$ is a graded object of $\mathcal{A}$ and that $d_0$ is compatible with the grading. Hence clearly $E_1$ is a graded object as well. But it turns out that the differential $d_1$ does not preserve this grading; instead it shifts the degree by $1$.
To work this out precisely, we define
\[ Z_ r^ p = \frac{F^ pK \cap d^{-1}(F^{p + r}K) + F^{p + 1}K}{F^{p + 1}K} \]
and
\[ B_ r^ p = \frac{F^ pK \cap d(F^{p - r + 1}K) + F^{p + 1}K}{F^{p + 1}K}. \]
This notation, although quite natural, seems to be different from the notation in most places in the literature. Perhaps it does not matter, since the literature does not seem to have a consistent choice of notation either. With these choices we see that $B_ r \subset E_0$, resp. $Z_ r \subset E_0$ (as defined in Section 12.22) is equal to $\bigoplus _ p B_ r^ p$, resp. $\bigoplus _ p Z_ r^ p$. Hence if we define
\[ E_ r^ p = Z_ r^ p/B_ r^ p \]
for $r \geq 0$ and $p \in \mathbf{Z}$, then we have $E_ r = \bigoplus _ p E_ r^ p$. We can define a differential $d_ r^ p : E_ r^ p \to E_ r^{p + r}$ by the rule
\[ z + F^{p + 1}K \longmapsto dz + F^{p + r + 1}K \]
where $z \in F^ pK \cap d^{-1}(F^{p + r}K)$.
Lemma 12.23.2. Let $\mathcal{A}$ be an abelian category. Let $(K, F, d)$ be a filtered differential object of $\mathcal{A}$. There is a spectral sequence $(E_ r, d_ r)_{r \geq 0}$ in $\text{Gr}(\mathcal{A})$ associated to $(K, F, d)$ such that $d_ r : E_ r \to E_ r[r]$ for all $r$ and such that the graded pieces $E_ r^ p$ and maps $d_ r^ p : E_ r^ p \to E_ r^{p + r}$ are as given above. Furthermore, $E_0^ p = \text{gr}^ p K$, $d_0^ p = \text{gr}^ p(d)$, and $E_1^ p = H(\text{gr}^ pK, d)$.
Proof. If $\mathcal{A}$ has countable direct sums and if countable direct sums are exact, then this follows from the discussion above. In general, we proceed as follows; we strongly suggest the reader skip this proof. Consider the object $A = (F^{p + 1}K)$ of $\text{Gr}(\mathcal{A})$, i.e., we put $F^{p + 1}K$ in degree $p$ (the funny shift in numbering to get numbering correct later on). We endow it with a differential $d$ by using $d$ on each component. Then $(A, d)$ is a differential object of $\text{Gr}(\mathcal{A})$. Consider the map
\[ \alpha : A \to A[-1] \]
which is given in degree $p$ by the inclusions $F^{p + 1}A \to F^ pA$. This is clearly an injective morphism of differential objects $\alpha : (A, d) \to (A, d)[-1]$. Hence, we can apply Remark 12.22.6 with $S = \text{id}$ and $T = [1]$. The corresponding spectral sequence $(E_ r, d_ r)_{r \geq 0}$ in $\text{Gr}(\mathcal{A})$ is the spectral sequence we are looking for. Let us unwind the definitions a bit. First of all we have $E_ r = (E_ r^ p)$ is an object of $\text{Gr}(\mathcal{A})$. Then, since $T^ rS = [r]$ we have $d_ r : E_ r \to E_ r[r]$ which means that $d_ r^ p : E_ r^ p \to E_ r^{p + r}$.
To see that the description of the graded pieces hold, we argue as above. Namely, first we have $E_0 = \mathop{\mathrm{Coker}}(\alpha : A \to A[-1])$ and by our choice of numbering above this gives $E_0^ p = \text{gr}^ pK$. The first differential is given by $d_0^ p = \text{gr}^ pd : E_0^ p \to E_0^ p$. Next, the description of the boundaries $B_ r$ and the cocycles $Z_ r$ in Remark 12.22.6 translates into a straightforward manner into the formulae for $Z_ r^ p$ and $B_ r^ p$ given above. $\square$
Lemma 12.23.3. Let $\mathcal{A}$ be an abelian category. Let $(K, F, d)$ be a filtered differential object of $\mathcal{A}$. The spectral sequence $(E_ r, d_ r)_{r \geq 0}$ associated to $(K, F, d)$ has equal to the boundary map in homology associated to the short exact sequence of differential objects
\[ d_1^ p : E_1^ p = H(\text{gr}^ pK) \longrightarrow H(\text{gr}^{p + 1}K) = E_1^{p + 1} \]
\[ 0 \to \text{gr}^{p + 1}K \to F^ pK/F^{p + 2}K \to \text{gr}^ pK \to 0. \]
Proof. This is clear from the formula for the differential $d_1^ p$ given just above Lemma 12.23.2. $\square$
Definition 12.23.4. Let $\mathcal{A}$ be an abelian category. Let $(K, F, d)$ be a filtered differential object of $\mathcal{A}$. The induced filtration on $H(K, d)$ is the filtration defined by $F^ pH(K, d) = \mathop{\mathrm{Im}}(H(F^ pK, d) \to H(K, d))$.
Writing out what this means we see that
\[ F^ pH(K, d) = \frac{\mathop{\mathrm{Ker}}(d) \cap F^ pK + \mathop{\mathrm{Im}}(d)}{\mathop{\mathrm{Im}}(d)} \]
and hence we see that
\[ \text{gr}^ p H(K) = \frac{\mathop{\mathrm{Ker}}(d) \cap F^ pK + \mathop{\mathrm{Im}}(d)}{\mathop{\mathrm{Ker}}(d) \cap F^{p + 1}K + \mathop{\mathrm{Im}}(d)} = \frac{\mathop{\mathrm{Ker}}(d) \cap F^ pK}{\mathop{\mathrm{Ker}}(d) \cap F^{p + 1}K + \mathop{\mathrm{Im}}(d) \cap F^ pK} \]
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. $\square$
Definition 12.23.6. Let $\mathcal{A}$ be an abelian category. Let $(K, F, d)$ be a filtered differential object of $\mathcal{A}$. We say the spectral sequence associated to $(K, F, d)$
weakly converges to $H(K)$ if $\text{gr}H(K) = E_{\infty }$ via Lemma 12.23.5,
abuts to $H(K)$ if it weakly converges to $H(K)$ and we have $\bigcap F^ pH(K) = 0$ and $\bigcup F^ pH(K) = H(K)$,
Unfortunately, it seems hard to find a consistent terminology for these notions in the literature.
Lemma 12.23.7. Let $\mathcal{A}$ be an abelian category. Let $(K, F, d)$ be a filtered differential object of $\mathcal{A}$. The associated spectral sequence
weakly converges to $H(K)$ if and only if for every $p \in \mathbf{Z}$ we have equality in equations (12.23.5.2) and (12.23.5.1),
abuts to $H(K)$ if and only if it weakly converges to $H(K)$ and $\bigcap _ p (\mathop{\mathrm{Ker}}(d) \cap F^ pK + \mathop{\mathrm{Im}}(d)) = \mathop{\mathrm{Im}}(d)$ and $\bigcup _ p (\mathop{\mathrm{Ker}}(d) \cap F^ pK + \mathop{\mathrm{Im}}(d)) = \mathop{\mathrm{Ker}}(d)$.
Proof. Immediate from the discussions above. $\square$
Post a comment
Your email address will not be published. Required fields are marked.
In your comment you can use Markdown and LaTeX style mathematics (enclose it like $\pi$). A preview option is available if you wish to see how it works out (just click on the eye in the toolbar).
All contributions are licensed under the GNU Free Documentation License.
Comments (5)
Comment #1035 by JuanPablo on
Comment #1036 by JuanPablo on
Comment #1044 by Johan on
Comment #2035 by Yu-Liang Huang on
Comment #2073 by Johan on