and
nonabelian homological algebra
Given a double complex $C_{\bullet, \bullet}$ (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 $Tot(C_{\bullet, \bullet})$.
This is the special case of the spectral sequence of a filtered complex with $Tot(C)_\bullet$ filtered by row-degree (or dually, by column-degree).
Let $C_{\bullet,\bullet}$ be a double complex. Its total complex $Tot C$ is given in degree $n$ by the direct sum
and the differential acts on elements $c \in C_{p,q}$ as
The horizontal filtration on $Tot C$ is the filtration $F_\bullet Tot C$ given in degree $n$ by the direct sum expression
Similarly the vertical filtration is given by
The (vertical/horizontal) spectral sequence of the double complex $C_{\bullet,\bullet}$ is the spectral sequence of a filtered complex for the filtered total complex from def. 1.
Let $\{E^r_{p,q}\}_{r,p,q}$ be the spectral sequence of a double complex $C_{\bullet, \bullet}$, according to def. 2, with respect to the horizontal filtration. Then the first few pages are for all $p,q \in \mathbb{Z}$ given by
$E^0_{p,q} \simeq C_{p,q}$;
$E^1_{p,q} \simeq H_q(C_{p, \bullet})$;
$E^2_{p,q} \simeq H_p(H^{vert}_q(C))$.
Moreover, if $C_{\bullet, \bullet}$ is concentrated in the first quadrant ($0 \leq p,q$), 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 $[c] \in E^1_{p,q}$ are given by $c \in C_{p,q}$ such that $\partial^{vert} c = 0$. It follows that $\partial^1 \colon E^1_{p,q} \to E^1_{p-1, q}$, which by the definition of a total complex acts as $\partial^{hor} \pm \partial^{vert}$, acts on these representatives just as $\partial^{hor}$ and this gives the second page
Finally, for $C_{\bullet, \bullet}$ concentrated in $0 \leq p,q$ the corresponding filtered chain complex $F_p Tot(C)_\bullet$ 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
or in section 25, lecture 9 of
Details are usually discussed for the more general case of a spectral sequence of a filtered complex.