Spectral Sequence of a filtration

Tags: #spectral-sequences #homological-algebra

Spectral Sequence of a filtration

Notes

Suppose we are given a chain complex \((C^{-}, d^{-})\) and we want to compute its homology. We assume that \(C^{<0} = 0\) (where we use homological indexing and \(d^{-}\) is of degree \(-1\)) along with an ascending, bounded filtration \(F^{-}C^{-}\) which can be written as

\begin{align*}0 =F^0C^{-}\subseteq F^1C^{-}\subseteq \ldots \subseteq F^{p-1}C^{-}\subseteq F^pC^{-}= C^{-}\end{align*}

and satisfies \(\bigcup_{i} F^i C^{-}= C^{-}\) and \(\bigcap_{i} C^{-}= \left\{{0}\right\}\).

Then take the associated graded complex, defined by

\begin{align*} G^i C^{-}= \frac{F^i C^{-}}{F^{i-1} C^{-}} \end{align*}

This yields a collection of short exact sequences of the form \begin{align*} 0 \to F^{i-1} C^{-}\to F^i C^{-}\to G^i C^{-}\to 0 \end{align*}

We now verify three facts:

  • Each \(F^i C^{-}\) is a chain complex (restrict the differential).
  • Each \(G^i C^{-}\) is a chain complex (differential well-defined on quotient).
  • \(F^{-}\) induces a filtration on \(H_*(C^{-})\), which we’ll denote \(F^{-}H_*(C^{-})\).

Given the induced filtration on homology, we can take its associated graded complex:

\begin{align*} F^i H_j(C^{-}) = \frac{?}{?} = \left\{{\alpha \in H_j(C^{-}) \mathrel{\Big|}\alpha = [x] \text{ for some } x \in F^iC^j}\right\} \end{align*}

Which yields a collection of short exact sequences \begin{align*} 0 \to F^{i-1}H_j(C^{-}) \to F^iH_j(C^{-}) \to G^iH_j(C^{-}) \to 0\end{align*}

And since \(F^pC^{-}= C^{-}\), we have \(F^pH_j(C^{-}) = H_j(C^{-})\). Assuming all sequences split and all extensions are unique, we can rewrite the left hand side:

\begin{align*} F^pH_j(C^{-}) &= F^{p-1}H_j(C^{-}) \oplus G^pH_j(C^{-}) \\ &= F^{p-2}H_j(C^{-}) \oplus G^{p-1}H_j(C^{-}) \oplus G^pH_j(C^{-}) \\ &\cdots \\ &= \bigoplus_{0\leq i \leq p}G^iH_j(C^{-}) \end{align*}

So if we are able to compute each \(G^iH_j(C^{-})\), we can recover the desired homology.

We proceed by computing \(H_j(G^i C^{-})\) instead, which we hope will be related to \(G^i H_j (C^{-})\).

By an earlier argument, we know that there exists induced differentials on the associated graded complex

\begin{align*} d_{i,j}: G^i C^j \to G^{i} C^{j-1} \end{align*}

We thus build the 0 page of our spectral sequence by defining

\begin{align*}E^0_{p,q} = G^p C^{p+q}\end{align*}

with a defined differential

\begin{align*}d^0_{i,j}: E^0_{i,j} \to E^0_{i-1, j}\end{align*}

and arranging these in columns, yielding the following situation

Now, since each \(G^i C^{-}\) is a chain complex, we can take the homology with respect to these differentials, so we define

\begin{align*} E^1_{i,j} = \frac {\ker\left( d^0_{i,j}: E^0_{i,j} \to E^0_{i-1, j} \right)} {\operatorname{im}\left(d^0_{i+1,j}: E^0_{i+1,j} \to E^0_{i,j} \right)} \coloneqq H_{i+j}(G^i C^{-}) \end{align*}

Which yields the following \(E^1\) page:

Which by definition is

We now claim that there is a differential

\begin{align*} d_1^{i,j}: H_n(G^i C^{-}) \to H_{n-1}(G^{i-1}C^{-}) \end{align*}

#spectral-sequences #homological-algebra