The spectral sequence of a filtered chain complex
is a tool for computing the chain homology of 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 by approximating cycles and boundaries of by their “-approximation”: an -almost cycle is a chain in filtering degree whose differential vanishes only up to terms that are steps lower in filtering degree, and an -almost boundary in filtering degree is a cycle that is the differential of a chain which may be (only) up to -degrees higher in filtering degree. The corresponding -almost homology of in filtering degree is the term of the spectral sequence.
If the filtering is bounded then -almost homology for sufficiently large (“-almost homology”) is 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 (it “collapses”), and so allows one to obtain the genuine homology from some finite -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
Via relative homology
Let be a ring and write Mod for its category of modules.
be a filtered chain complex in , with associated graded complex denoted .
-Almost cycles and boundaries
Given a filtered chain complex as above we say for all that
is the module of -chains or of -chains in filtering degree ;
is the module of -almost -cycles (the -chains whose differentials vanish modulo terms of filtering degree );
is the module of -almost -boundaries.
Similarly we set
From this definition we immediately have that the differentials restrict to the -almost cycles as follows:
The differentials of restrict on -almost cycles to morphisms of the form
These are still differentials: .
By the very definition of it consists of elements in filtering degree on which decreases the filtering degree to . Also by definition of differential on a chain complex, decreases the actual degree by one. This explains that restricted to lands in .
Now the image consists indeed of actual boundaries, not just -almost boundaries. But since actual boundaries are in particular -almost boundaries, we may take the codomain to be .
We have a sequence of canonical inclusions
The -almost cycles are the -kernel inside the -almost cycles:
An element represents
an element in if
an element in if even .
The second condition is equivalent to representing the 0-element in the quotient . But this is in turn equivalent to being 0 in .
-Almost homology groups: the spectral sequence
Let be a filtered chain complex as above.
For define the -almost -chain homology of the filtered complex to be the quotient of the -almost -cycles by the -almost -boundaries, def. 1:
By prop. 1 the differentials of restrict on the -almost homology groups to maps
Definition 2 indeed gives a spectral sequence in that is indeed the -chain homology of , i.e.
Via exact couples
The whole of the spectral sequence can be defined as the spectral sequence of the exact couple
and where is the cohomology morphism induced by the inclusion of chain complexes
and is the total cohomology of the associated bigraded complex.
At every stage we have a new family of long exact sequences.
In low-degree pages
We characterize the value of the spectral sequence , def. 2 for low values of and, below in prop. 8, for .
For the spectral seuence of a filtered complex from def. 2, the first pages have the following form:
is the associated p-graded piece of ;
is the chain homology of the associated p-graded complex .
For def. 2 restricts to
because for we automatically also have since the differential respects the filtering degree by assumption.
For def. 2 gives
Let be a spectral sequence such that for each there is such that for all we have
Then one says that
the bigraded object
is the limit term of the spectral sequence;
- the spectral sequence abuts to .
If for a spectral sequence there is such that all differentials on pages after vanish, , then is limit term for the spectral sequence. One says in this cases that the spectral sequence collapses at .
By the defining relation
the spectral sequence becomes constant in from on if all the differentials vanish, so that for all .
If for a spectral sequence there is such that the th page is concentrated in a single row or a single column, then the the spectral sequence degenerates on this page, 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 the differentials of the spectral sequence
have domain and codomain necessarily in different rows an columns (while for both are in the same row and for both coincide). Therefore if all but one row or column vanish, then all these differentials vanish.
A spectral sequence is said to converge to a graded object with filtering , traditionally denoted
if the associated graded complex of is the limit term of , def. 3:
A spectral sequence is called a bounded spectral sequence if for all the number of non-vanishing terms of the form is finite.
A spectral sequence is called
a first quadrant spectral sequence if all terms except possibly for vanish;
a third quadrant spectral sequence if all terms except possibly for vanish.
Such spectral sequences are bounded, def. 5.
A bounded spectral sequence, def. 5, has a limit term, def. 3.
First notice that if a spectral sequence has at most non-vanishing terms of total degree on page , 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 there is such that for all and all . Similarly there is such for all and all .
We claim then that the limit term of the bounded spectral sequence is in position given by the value for
This is because for such we have
because , and hence the kernel vanishes;
because , and hence the image vanishes.
Convergence of bounded filtrations
A filtration on a chain complex is called a bounded filtration if for all there is such that
The spectral sequence of a complex with bounded filtration, def. 6, has a limit term, def. 3.
The spectral sequence of a filtered complex is by def. 2 at a quotient of a subobject of . By def. 6 therefore there are for each finitely many non-vanishing terms of the form . Therefore the spectral sequence is bounded, def. 5 and hence has a limit term by prop. 6.
If Top is a CW complex with cell filtration , then the induced filtering
on its singular chain complex yields a first-quadrant spectral sequence, example 3. Therefore it has a limit term.
Before saying 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 a filtered complex, write for
This defines a filtering of the homology, regarded as a graded object.
If the spectral sequence of a filtered complex , def. 6 has a limit term, def. 3, then it converges, def. 4, to the chain homology of the complex:
Hence for each there is such that
with the filtering on the right as in def. 7.
By assumption, there is for each an such that for all the -almost cycles and -almost boundaries, def. 1, in are the ordinary cycles and boundaries. Therefore for def. 2 gives .
Via exact couples
It is instructive to note that in the th derived exact couple , the hidden part is the submodule of , as it meets representable by elements of ; that is, we may sensibly call it
Separating the grades, the exactness of the couple at then says
One can see this as converging (if it sensibly converges) to either a subquotient of or to a submodule . Taking the latter interpretation, we hope to find in the limiting case exact sequences
At this stage one can check that the morphisms are indeed definable, and in fact injective, so that whatever should be, the morphism is null; that is, our long-exact sequence breaks up into the short exact sequences
In summary, if the spectral sequence converges in a sensible way to the correct thing , then that correct thing is also the associated graded module of the filtration of induced by the filtration of .
The special case where the filtering has just length one is that where we simply have a sub-complex and want to compute the homology of from that of and .
This case is easily solved by elementary means and it serves as an instructive blueprint for the general case.
Given a sub-chain complex , consider the following constructions
Consider the short exact sequence
Its long exact sequence in homology contains the connecting homomorphism
Then is sits in the short exact sequence
Homology of a tensor product of chain complexes
Consider two chain complexes of vector spaces over a field , both in non-negative degree.
Their tensor product of chain complexes is
with differential on homogenous elements
We may compute the homology of by a spectral sequence as follows.
Define a filtration on by
This means that the associated graded object is simply
The differential on this is . Hence the universal coefficient theorem gives
The next differential is . Since is assumed to be a field we have thus
Therefore every element in is represented by a tensor product of a -cycle with a -cycle and is hence itself a -cycle. Since the differentials in the spectral sequence all come from the differential on , this means that all higher differentials vanish, and so the sequence collapses on the -page.
The convergence of the spectral sequence to the the homology of thus says that this is given by
Homology of the total complex of a double complex
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.
Singular homology of a CW-complex
Let be a CW-complex equipped explicitly with the structure of a filtered topological space . This induces on the singular homology complex the structure of a filtered chain complex by
We discuss how the corresponding spectral sequence shows that the singular homology of 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 is the cell decomposition of a cell complex we have
The chain homology of
is the cellular chain homology . One finds that
Since this is concentrated in the -row all higher- differentials vanish.
Serre’s spectral sequence of a fibration
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. Let
be a Serre fibration of pointed topological spaces in which is a connected CW-complex. Then the -skeleta of naturally give filtered-space structures to both and :
and in turn induce filtrations of the singular chain complex of .
The homology Serre spectral sequence for the fibration is essentially that of this filtered complex.
It is straight-forward to show that the pair is -connected, and in particular the relative homology vanishes for ; 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 of the homology Serre spectral sequence is given by
the homology of with coefficients in the local system defined by the action of on . In the special case that is simply connected, these local systems are canonically equivalent to , the homology of the fiber over the basepoint.
Lecture notes include
and section 3 of
For further references see those listed at spectral sequence, for instance section 5 of