Given a double complex (in some abelian category), there is a spectral sequence whose second page is the “naïve double cohomology”
and which converges, under mild conditions, to the correct chain homology of the total complex .
This is the special case of the spectral sequence of a filtered complex with filtered by row-degree (or dually, by column-degree).
Let be a double complex. Its total complex is given in degree by the direct sum
and the differential acts on elements as
The horizontal filtration on is the filtration given in degree by the direct sum expression
Similarly the vertical filtration is given by
Let be the spectral sequence of a double complex , according to def. 2, with respect to the horizontal filtration. Then the first few pages are for all given by
Moreover, if is concentrated in the first quadrant (), then the spectral sequence converges to the chain homology of the total complex:
This is a matter of unwinding the definition, using the general properties of spectral sequences of a filtered complex – in low degree pages. We display equations for the horizontal filtering, the other case works analogously.
The 0th page is by definition the associated graded piece
The first page is the chain homology of the associated graded chain complex:
In particular this means that representatives of are given by such that . It follows that , which by the definition of a total complex acts as , acts on these representatives just as and this gives the second page
Finally, for concentrated in the corresponding filtered chain complex is a non-negatively graded chain complex with filtration bounded below. Therefore the spectral sequence converges as claimed by the general discussion at spectral sequence of a filtered complex - convergence.
A plethora of types of spectral sequences are special cases of the spectral sequence of a double complex. For the moment see at spectral sequence for a list.
Dedicated discussion of the case of spectral sequences of double complexes is for instance in
- Ravi Vakil, Spectral Sequences: Friend or Foe? (pdf)
or in section 25, lecture 9 of
Details are usually discussed for the more general case of a spectral sequence of a filtered complex.