sequences constructed with this method Serre spectral sequence - used to compute (co)homology of a fibration Atiyah–Hirzebruch spectral sequence - used to compute (co)homology of extraordinary cohomology theories, such as K-theory Bockstein spectral sequence. Spectral sequences of filtered complexes The spectral sequence of a filtered complex A very common type of spectral sequence comes from a filtered cochain complex, as it naturally induces a bigraded object. Consider a cochain complex together with a descending filtration, . We require that the boundary map is compatible with the filtration, i.e. , and that the filtration is exhaustive, that is, the union of the set of all is the entire chain complex . Then there exists a spectral sequence with and . Later, we will also assume that the filtration is Hausdorff or separated, that is, the intersection of the set of all is zero. The filtration is useful because it gives a measure of nearness to zero: As p increases, 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 . Construction has only a single grading and a filtration, so we first construct a doubly graded object for the first page of the spectral sequence. 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 step: Since we assumed that the boundary map was compatible with the filtration, is a doubly graded object and there is a natural doubly graded boundary map on . To get , we take the homology of . Notice that and can be written as the images in of and that we then have are exactly the elements which the differential pushes up one level in the filtration, and are exactly the image of the elements which the differential pushes up zero levels in the filtration. This suggests that we should choose to be the elements which the differential pushes up r levels in the filtration and to be image of the elements which the differential pushes up r-1 levels in the filtration. In other words, the spectral sequence should satisfy and we should have the