and
nonabelian homological algebra
The spectral sequence of a filtered chain complex
is a tool for computing the chain homology of $C_\bullet$ from the chain homologies of the associated graded objects
which is in general simpler. This is a special case of the spectral sequence of a filtered stable homotopy type.
The sequence asymptotes to the homology of $C_\bullet$ by approximating cycles and boundaries of $C$ by their “$r$-approximation”: an $r$-almost cycle is a chain in filtering degree $p$ whose differential vanishes only up to terms that are $r$ steps lower in filtering degree, and an $r$-almost boundary in filtering degree $p$ is a cycle that is the differential of a chain which may be (only) up to $r$-degrees higher in filtering degree. The corresponding $r$-almost homology of $C_\bullet$ in filtering degree $p$ is the term $E^r_{p,\bullet}$ of the spectral sequence.
If the filtering is bounded then $r$-almost homology for sufficiently large $r$ (“$\infty$-almost homology”) is the clearly the genuine homology, and so the spectral sequence converges to the correct homology. But the point is that typically it reaches the correct value already at some low finite degree $r$ (it “collapses”), and so allows one to obtain the genuine homology from some finite $r$-almost homology.
One may also regard the spectral sequence of a filtered complex as a tool for organizing data derivable from the families of long exact sequence in homology
which are induced by the short exact sequences
coming from the filtering.
We give the definition
and
Let $R$ be a ring and write $\mathcal{A} = R$Mod for its category of modules.
Let
be a filtered chain complex in $\mathcal{A}$, with associated graded complex denoted $G_\bullet C_\bullet$.
In more detail this means that
$[\cdots \stackrel{\partial_{n}}{\to} C_n \stackrel{\partial_{n-1}}{\to}] C_{n-1} \to \cdots]$ is a chain complex, hence $\{C_n\}$ are objects in $\mathcal{A}$ ($R$-modules) and $\{\partial_n\}$ are morphisms (module homomorphisms) with $\partial_n \circ \partial_{n+1} = 0$;
For each $n \in \mathbb{Z}$ there is a filtering $F_\bullet C_n$ on $C_n$ and all these filterings are compatible with the differentials in that
The grading associated to the filtering is such that the $p$-graded elements are those in the quotient
Since the differentials respect the grading we have chain complexes $G_p C_\bullet$ in each filtering degree $p$.
We use element-notation in the following as if $\mathcal{A}$ were a category of modules.
Given a filtered chain complex $F_\bullet C_\bullet$ as above we say for all $r, p, q \in \mathbb{Z}$ that
$G_p C_{p+q}$ is the module of $(p,q)$-chains or of $(p+q)$-chains in filtering degree $p$;
$\begin{aligned} Z^r_{p,q} & \coloneqq \left\{ c \in G_p C_{p+q} | \partial c = 0 \, mod F_{p-r} C_{\bullet} \right\} \\ & = \left\{ c \in F_p C_{p+q} | \partial(c) \in F_{p-r} C_{p+q-1} \right\}/ F_{p-1}C_{p+q} \end{aligned}$
is the module of $r$-almost $(p,q)$-cycles (the $(p+q)$-chains whose differential vanishes modulo terms of filtering degree $p-r$);
$B^{r}_{p,q} \coloneqq \partial(F_{p+r-1} C_{p+q+1}) \,,$
is the module of $r$-almost $(p,q)$-boundaries.
Similarly we set
From this definition we immediately have that the differentials $\partial \colon C_{p+q} \to C_{p+q-1}$ restrict to the $r$-almost cycles as follows:
The differentials of $C_\bullet$ restrict on $r$-almost cycles to morphisms of the form
These are still differentials: $\partial^2 = 0$.
By the very definition of $Z^r_{p,q}$ it consists of elements in filtering degree $p$ on which $\partial$ decreases the filtering degree to $p-r$. Also by definition of differential on a chain complex, $\partial$ decreases the actual degree $p+q$ by one. This explains that $\partial$ restricted to $Z^r_{p,q}$ lands in $Z^\bullet_{p-r,q+r-1}$.
Now the image constists indeed of actual boundaries, not just $r$-almost boundaries. But since actual boundaries are in particular $r$-almost boundaries, we may take the codomain to be $Z^r_{p-r,q+r-1}$.
We have a sequence of canonical inclusions
The $(r+1)$-almost cycles are the $\partial^r$-kernel inside the $r$-almost cycles:
An element $c \in F_p C_{p+q}$ represents
an element in $Z^r_{p,q}$ if $\partial c \in F_{p-r} C_{p+q-1}$
an element in $Z^{r+1}_{p,q}$ if even $\partial c \in F_{p-r-1} C_{p+q-1} \hookrightarrow F_{p-r} C_{p+q-1}$.
The second condition is equivalent to $\partial c$ representing the 0-element in the quotient $F_{p-r}C_{p+q-1}/ F_{p-r-1}C_{p+q-1}$. But this is in turn equivalent to $\partial c$ being 0 in $Z^r_{p-r,q+r-1} \subset F_{p-r} C_{p+q-1} / F_{p-r-1} C_{p+q-1}$.
Let $F_\bullet C_\bullet$ be a filtered chain complex as above.
For $r, p, q \in \mathbb{Z}$ define the $r$-almost $(p,q)$-chain homology of the filtered complex to be the quotient of the $r$-almost $(p,q)$-cycles by the $r$-almost $(p,q)$-boundaries, def. 1:
By prop. 1 the differentials of $C_\bullet$ restrict on the $r$-almost homology groups to maps
Definition 2 indeed gives a spectral sequence in that $E^{r+1}_{\bullet, \bullet}$ is indeed the $\partial^r$-chain homology of $E^r_{\bullet, \bullet}$, i.e.
By prop. 3.
The whole of the spectral sequence can be defined as the spectral sequence of the exact couple
where
$D \coloneqq \bigoplus_i H^{\bullet}(F_i)$
and where $\varphi$ is the cohomology morphism induced by the inclusion of chain complexes $F_i\to F_{i+1}$
and $E \coloneqq \bigoplus_i H^\bullet(F_i/F_{i-1})$ is the total cohomology of the associated bigraded complex.
At every stage we have a new family of long exact sequences
We characterize the value of the spectral sequence $E^r_{p,q}$, def. 2 for low values of $r$ and, below in prop. \ref{ConvergingToGenuineHomology}, for $r \to \infty$.
We have
$E^0_{p,q} = G_p C_{p+q} = F_p C_{p+q} / F_{p-1} C_{p+q}$
is the associated p-graded piece of $C_{p+q}$;
$E^1_{p,q} = H_{p+q}(G_p C_\bullet)$
is the chain homology of the associated p-graded complex $G_p C_\bullet$.
For $r = 0$ def. 2 restricts to
because for $c \in F_p C_{p+q}$ we automatically also have $\partial c \in F_p C_{p+q}$ since the differential respects the filtering degree by assumption.
For $r = 1$ def. 2 gives
There is, in general, a decisive difference between the homology of the associated graded complex $H_{p+q}(G_p C_\bullet)$ and the associated graded piece of the genuine homology $G_p H_{p+q}(C_\bullet)$: in the former the differentials of cycles are required to vanish only up to terms in lower degree, but in the latter they are required to vanish genuinely. The latter expression is instead the value of the spectral sequence for $r \to \infty$, see prop. \ref{ConvergingToGenuineHomology} below.
Let $\{E^r_{p,q}\}_{r,p,q}$ be a spectral sequence such that for each $p,q$ there is $r(p,q)$ such that for all $r \geq r(p,q)$ we have
Then one says that
the bigraded object
is the limit term of the spectral sequence;
If for a spectral sequence there is $r_s$ such that all differentials on pages after $r_s$ vanish, $\partial^{r \geq r_s} = 0$, then $\{E^{r_s}\}_{p,q}$ is limit term for the spectral sequence. One says in this cases that the spectral sequence collapses at $r_s$.
By the defining relation
the spectral sequence becomes constant in $r$ from $r_s$ on if all the differentials vanish, so that $ker(\partial^r_{p,q}) = E^r_{p,q}$ for all $p,q$.
If for a spectral sequence $\{E^r_{p,q}\}_{r,p,q}$ there is $r_s \geq 2$ such that the $r_s$th page is concentrated in a single row or a single column, then the the spectral sequence degenerates on this pages, example 1, hence this page is a limit term, def. 3. One says in this case that the spectral sequence collapses on this page.
For $r \geq 2$ the differentials of the spectral sequence
have domain and codomain necessarily in different rows an columns (while for $r = 1$ both are in the same row and for $r = 0$ both coincide). Therefore if all but one row or column vanish, then all these differentials vanish.
A spectral sequence $\{E^r_{p,q}\}_{r,p,q}$ is said to converge to a graded object $H_\bullet$ with filtering $F_\bullet H_\bullet$, traditionally denoted
if the associated graded complex $\{G_p H_{p+q}\}_{p,q} \coloneqq \{F_p H_{p+q} / F_{p-1} H_{p+q}\}$ of $H$ is the limit term of $E$, def. 3:
In practice spectral sequences are often referred to via their first non-trivial page, often also the page at which it collapses, def. 2, oftne the second page. Then one often uses notation such as
to be read as “There is a spectral sequence whose second page is as shown on the left and which converges to a filtered object as shown on the right.”
A spectral sequence $\{E^r_{p,q}\}$ is called a bounded spectral sequence if for all $n,r \in \mathbb{Z}$ the number of non-vanishing terms of the form $E^r_{k,n-k}$ is finite.
A spectral sequence $\{E^r_{p,q}\}$ is called
a first quadrant spectral sequence if all terms except possibly for $p,q \geq 0$ vanish;
a third quadrant spectral sequence** if all terms except possibly for $p,q \leq 0$ vanish.
Such spectral sequences are bounded, def. 5.
First notice that if a spectral sequence has at most $N$ non-vanishing terms of total degree $n$ on page $r$, then all the following pages have at most at these positions non-vanishing terms, too, since these are the homologies of the previous terms.
Therefore for a bounded spectral sequence for each $n$ there is $L(n) \in \mathbb{Z}$ such that $E^r_{p,n-p} = 0$ for all $p \leq L(n)$ and all $r$. Similarly there is $T(n) \in \mathbb{Z}$ such $E^r_{n-q,q} = 0$ for all $q \leq T(n)$ and all $r$.
We claim then that the limit term of the bounded spectral sequence is in position $(p,q)$ given by the value $E^r_{p,q}$ for
This is because for such $r$ we have
$E^r_{p-r, q+r-1} = 0$ because $p-r \lt L(p+q-1)$, and hence the kernel $ker(\partial^r_{p-r,q+r-1}) = 0$ vanishes;
$E^r_{p+r, q-r+1} = 0$ because $q-r + 1 \lt T(p+q+1)$, and hence the image $im(\partial^r_{p,q}) = 0$ vanishes.
Therefore
A filtration $F_\bullet C_\bullet$ on a chain complex $C_\bullet$ is called a bounded filtration if for all $n \in \mathbb{Z}$ there is $L(n), T(n) \in \mathbb{Z}$ such that
The spectral sequence of a filtered complex $F_\bullet C_\bullet$ is by def. 2 at $E^r_{p,q}$ a quotient of a subobject of $F_p C_{p+q}$. By def. 6 therefore there are for each $n,r \in \mathbb{Z}$ finitely many non-vanishing terms of the form $E^r_{p,n-p}$. Therefore the spectral sequence is bounded, def. 5 and hence has a limit term by prop. 6.
If $X \in$ Top is a CW complex with cell filtration $X_0 \hookrightarrow X_1 \hookrightarrow \cdots \hookrightarrow X$, then the induced filtering
on its singular chain complex $C_\bullet(X)$ yields a first-quadrant spectral sequence, example 3. Therefore it has a limit term.
Before saying what that the spectral sequence of a filtered complex converges to the homology of that complex, we need to be careful about what the filtering is on that homology:
For $F_\bullet C_\bullet$ a filtered complex, write for $p \in \mathbb{Z}$
This defines a filtering $F_\bullet H_\bullet(C)$ of the homology, regarded as a graded object.
If the spectral sequence of a filtered complex $F_\bullet C_\bullet$, def. 6 has a limit term, def. 3, then it converges, def. 4, to the chain homology of this complex:
Hence for each $p,q$ there is $r(p,q)$ such that
with the filtering on the right as in def. 7.
By assumption, there is for each $p,q$ an $r(p,q)$ such that for all $r \geq r(p,q)$ the $r$-almost cycles and $r$-almost boundaries, def. 1, in $F_p C_{p+q}$ are the ordinary cycles and boundaries. Therefore for $r \geq r(p,q)$ def. 2 gives $E^r_{p,q} \simeq G_p H_{p+q}(C)$.
It is instructive to note that in the $n$th derived exact couple $\varphi^n D\to E_{(n)} \to \varphi^n D\to{}$, the hidden part $\varphi^n D$ is the submodule $D_{(n)}$ of $\bigoplus_{i} H(F_{i+n})$, as it meets $H(F_{i+n})$ representable by elements of $F_i$; that is, we may sensibly call it
Separating the grades, the exactness of the couple at $E_{(n)}$ then says
One can see this as converging (if it sensibly converges) to either a subquotient of $F_i$ or to a submodule $F_i H^\bullet(C) \lt H^\bullet(C)$. Taking the latter interpretation, we hope to find in the limiting case exact sequences
At this stage one can check that the morphisms $F_i H^j(C)\to F_{i+1} H^j(C)$ are indeed definable, and in fact injective, so that whatever $E_{(\infty)}$ should be, the morphism $E_{(\infty)}^{i\dots}\to F_i H^{j+1}$ is null; that is, our long-exact sequence breaks up into the short exact sequences
In summary, if the spectral sequence $E_{(n)}$ converges in a sensible way to the correct thing $E_{(\infty)}$, then that correct thing is also the associated graded module of the filtration of $H^\bullet(C)$ induced by the filtration of $C$.
The special case where the filtering has just length one is that where we simply have a sub-complex $C^{(1)}_\bullet \hookrightarrow C_\bullet$ and want to compute the homology of $C_\bullet$ from that of $C^{(1)}_\bullet$ and $C_\bullet/C^{(1)}_\bullet$.
This case is easily solved by elementary means and it serves as an instructive blueprint for the general case.
Given a sub-chain complex $C^{(1)}_\bullet \hookrightarrow C_\bullet$, consider the following constructions
Consider the short exact sequence
Its long exact sequence in homology contains the connecting homomorphism
Define
$G_1 H_\bullet \coloneqq ker \delta$
$G_2 H_\bullet \coloneqq coker \delta$.
Then $H_\bullet(C_\bullet)$ is sits in the short exact sequence
(…)
Consider two chain complexes $C_\bullet, C'_\bullet$ of vector spaces over a field $k$, both in non-negative degree.
Their tensor product of chain complexes is
with differential on homogenous elements
We may compute the chain homology of $C \otimes C'$ by a filter spectral sequence as follows.
Define a filtration on $C \otimes C'$ by
This means that the associated graded object is simply
The differential on this is $\partial_{r = 0} = (-1)^p id_{C} \otimes \partial'$. Hence the universal coefficient theorem gives
The next differential is $\partial_1 = \partial \otimes id_{C'}$ Since $k$ is assumed to be a field we have thus
Therefore every element in $E^2_{p,q}$ is represented by a tensor product of a $C$-cycle with a $C'$-cycle and is hence itself a $(C \otimes C')$-cycle. Since the differentials in the spectral sequence all come from the differential on $C \otimes C'$, this means that all higher differentials vanish, and so the sequence collapses on the $E^2$-page.
The convergence of the spectral sequence to the the homology of $C \otimes C'$ thus says that this is given by
The total complex of a double complex is naturally filtered either by either row-degree of column-degree. The corresponding filtering spectral sequence converges under good conditions to the homology of the total complex. See at spectral sequence of a double complex.
Let $X \in Top$ be a CW-complex equipped explicitly with the structure of a filtered topological space $X^0 \hookrightarrow \cdots \hookrightarrow X^n \hookrightarrow \cdots X$. This induces on the singular homology complex $C_\bullet(X)$ the structure of a filtered chain complex by
We discuss how the corresponding spectral sequence shows that the singular homology of $X$ coincides with the cellular homology of the filtering.
The associated graded object is
The chain homology of the associated graded chain complex is the relative homology
Now by assumption that $X^\bullet$ is the cell decomposition of a cell complex we have
The chain homology of
is the cellular chain homology $H^{cell}_p(X)$. One finds that
Since this is concentrated in the $q = 0$-row all higher-$r$ differentials vanish.
Hence $H_p(X) \simeq H^{cell}_p(X)$.
The generalization of this argument from ordinary homology to generalized homology is given by the Atiyah–Hirzebruch spectral sequence.
We indicate the Leray-Serre spectral sequence of a Serre fibration as a special case of the filtering spectral sequence. For more discussion see there.
be a Serre fibration of pointed topological spaces in which $B$ is a connected CW-complex. Then the $k$-skeleta of $B$ naturally give filtered-space structures to both $B$ and $X$:
and in turn induce filtrations of the singular chain complex of $X$.
The homology Serre spectral sequence for the fibration is essentially that of this filtered complex.
It is straight-forward to show that the pair $(X_{k+1};X_k)$ is $k$-connected, and in particular the relative homology $H_i(X_{k+1};X_k)$ vanishes for $i\leq k$; this ensures that the spectral sequence is 1st/3rd quadrant (And nota bene: this is also a handy way to remember what the bigrading actually is).
There is also an important result about the second page of this spectral sequence
The page $E^{(2)}$ of the homology Serre spectral sequence is given by
the homology of $B$ with coefficients in the local system defined by the action of $\Pi_1 (B)$ on $H_q(A|_{b})$. In the special case that $B$ is simply connected, these local systems are canonically equivalent to $H_q(A)$, the homology of the fiber over the basepoint.
A detailed account is in
Lecture notes include
and section 3 of
For further references see those listed at spectral sequence, for instance section 5 of