nLab spectral sequence of a double complex

Idea

Given a double complex $C_{\bullet, \bullet}$ (in some abelian category), there is a spectral sequence whose second page is the “naïve double cohomology”

$$E^2_{p,q} = H_p^{hor}(H_q^{vert}(C))$$

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).

Definition

Let $C_{\bullet,\bullet}$ be a double complex. Its total complex $Tot C$ is given in degree $n$ by the direct sum

$$(Tot C)_n = \oplus_{p+q = n} C_{p,q}$$

and the differential acts on elements $c \in C_{p,q}$ as

$$\partial^{Tot} \colon c \mapsto \partial^{hor} c + (-1)^{p} \partial^{ver} c \,.$$

Definition

The horizontal filtration on $Tot C$ is the filtration $F_\bullet Tot C$ given in degree $n$ by the direct sum expression

$$F^{hor}_p (Tot C)_n \coloneqq \oplus_{{n_1+n_2 = n} \atop {n_1 \leq p} } C_{n_1,n_2} \,.$$

Similarly the vertical filtration is given by

$$F^{vert}_p (Tot C)_n \coloneqq \oplus_{{n_1+n_2 = n} \atop {n_2 \leq p} } C_{n_1,n_2} \,.$$

Definition

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. .

Properties

Low-degree pages

Proposition

Let $\{E^r_{p,q}\}_{r,p,q}$ be the spectral sequence of a double complex $C_{\bullet, \bullet}$, according to def. , 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:

$$E^\infty_{p,q} \simeq G_p H_{p+q}(Tot(C)_\bullet) \,.$$

Proof

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

$$\begin{aligned} E^0_{p,q} & \coloneqq G_p Tot(C)_{p+q} \\ & \coloneqq F_p Tot(C)_{p+q} / F_{p-1} Tot(C)_{p+q} \\ & \coloneqq \frac{ \underset{ {n_1 + n_2 = p+q} \atop {n_1 \leq p} }{\oplus} C_{n_1, n_2} } { \underset{ {n_1 + n_2 = p+q} \atop {n_1 \lt p} }{\oplus} C_{n_1, n_2} } \\ & \simeq C_{p,q} \,. \end{aligned}$$

The first page is the chain homology of the associated graded chain complex:

$$\begin{aligned} E^1_{p,q} & \simeq H_{p+q}(G_p Tot(C)_\bullet) \\ & \simeq H_{p+q}( C_{p,\bullet} ) \\ & \simeq H_q(C_{p, \bullet}) \end{aligned} \,.$$

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

$$E^2_{p,q} \simeq ker(\partial^1_{p-1,q})/im(\partial^1_{p,q}) \simeq H_p(H_q^{vert}(C_{\bullet, \bullet})) \,.$$

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.

Examples

A plethora of types of spectral sequences are special cases of the spectral sequence of a double complex, for instance

• Frölicher spectral sequence.

For the moment see at spectral sequence for a list.

References

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

• Rick Jardine, Lectures on Homotopy Theory (pdf)

Details are usually discussed for the more general case of a spectral sequence of a filtered complex.