Spectral Sequences - Examples of Spectral Sequences - The Spectral Sequence of A Filtered Complex

The Spectral Sequence of A Filtered Complex

A very common type of spectral sequence comes from a filtered cochain complex. This is a cochain complex C• together with a set of subcomplexes FpC•, where p ranges across all integers. (In practice, p is usually bounded on one side.) We require that the boundary map is compatible with the filtration; this means that d(FpCn) ⊆ FpCn+1. We assume that the filtration is descending, i.e., FpC•Fp+1C•. We will number the terms of the cochain complex by n. Later, we will also assume that the filtration is Hausdorff or separated, that is, the intersection of the set of all FpC• is zero, and that the filtration is exhaustive, that is, the union of the set of all FpC• is the entire chain complex C•.

The filtration is useful because it gives a measure of nearness to zero: As p increases, FpC• gets closer and closer to zero. We will construct a spectral sequence from this filtration where coboundaries and cocycles in later sheets get closer and closer to coboundaries and cocycles in the original complex. This spectral sequence is doubly graded by the filtration degree p and the complementary degree q = np. (The complementary degree is often a more convenient index than the total degree n. For example, this is true of the spectral sequence of a double complex, explained below.)

We will construct this spectral sequence by hand. C• has only a single grading and a filtration, so we first construct a doubly graded object from C•. To get the second grading, we will take the associated graded object with respect to the filtration. We will write it in an unusual way which will be justified at the E1 step:

Since we assumed that the boundary map was compatible with the filtration, E0 is a doubly graded object and there is a natural doubly graded boundary map d0 on E0. To get E1, we take the homology of E0.

Notice that and can be written as the images in of

and that we then have

is exactly the stuff which the differential pushes up one level in the filtration, and is exactly the image of the stuff which the differential pushes up zero levels in the filtration. This suggests that we should choose to be the stuff which the differential pushes up r levels in the filtration and to be image of the stuff which the differential pushes up r-1 levels in the filtration. In other words, the spectral sequence should satisfy

and we should have the relationship

For this to make sense, we must find a differential dr on each Er and verify that it leads to homology isomorphic to Er+1. The differential

is defined by restricting the original differential d defined on to the subobject .

It is straightforward to check that the homology of Er with respect to this differential is Er+1, so this gives a spectral sequence. Unfortunately, the differential is not very explicit. Determining differentials or finding ways to work around them is one of the main challenges to successfully applying a spectral sequence.

Read more about this topic:  Spectral Sequences, Examples of Spectral Sequences

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