# nLab spectral sequence of a double complex

### Context

#### Homological algebra

homological algebra

and

nonabelian homological algebra

diagram chasing

# Contents

## Idea

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

${E}_{p,q}^{2}={H}_{p}^{\mathrm{hor}}\left({H}_{q}^{\mathrm{vert}}\left(C\right)\right)$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 $\mathrm{Tot}\left({C}_{•,•}\right)$.

This is the special case of the spectral sequence of a filtered complex with $\mathrm{Tot}\left(C{\right)}_{•}$ filtered by row-degree (or dually, by column-degree).

## Definition

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

$\left(\mathrm{Tot}C{\right)}_{n}={\oplus }_{p+q=n}{C}_{p,q}$(Tot C)_n = \oplus_{p+q = n} C_{p,q}

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

${\partial }^{\mathrm{Tot}}:c↦{\partial }^{\mathrm{hor}}c+\left(-1{\right)}^{p}{\partial }^{\mathrm{ver}}c\phantom{\rule{thinmathspace}{0ex}}.$\partial^{Tot} \colon c \mapsto \partial^{hor} c + (-1)^{p} \partial^{ver} c \,.
###### Definition

The horizontal filtration on $\mathrm{Tot}C$ is the filtration ${F}_{•}\mathrm{Tot}C$ given in degree $n$ by the direct sum expression

${F}_{p}^{\mathrm{hor}}\left(\mathrm{Tot}C{\right)}_{n}≔{\underset{\genfrac{}{}{0}{}{{n}_{1}+{n}_{2}=n\oplus }{{n}_{1}\le p}}{C}}_{{n}_{1},{n}_{2}}\phantom{\rule{thinmathspace}{0ex}}.$F^{hor}_p (Tot C)_n \coloneqq \underset{{n_1+n_2 = n}{\oplus} \atop {n_1 \leq p} } C_{n_1,n_2} \,.

Similarly the vertical filtration is given by

${F}_{p}^{\mathrm{vert}}\left(\mathrm{Tot}C{\right)}_{n}≔{\oplus }_{\genfrac{}{}{0}{}{{n}_{1}+{n}_{2}=n}{{n}_{2}\le p}}{C}_{{n}_{1},{n}_{2}}\phantom{\rule{thinmathspace}{0ex}}.$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 ${C}_{•,•}$ is the spectral sequence of a filtered complex for the filtered total complex from def. 1.

## Properties

### Low-degree pages

###### Proposition

Let $\left\{{E}_{p,q}^{r}{\right\}}_{r,p,q}$ be the spectral sequence of a double complex ${C}_{•,•}$, according to def. 2, with respect to the horizontal filtration. Then the first few pages are for all $p,q\in ℤ$ given by

• ${E}_{p,q}^{0}\simeq {C}_{p,q}$;

• ${E}_{p,q}^{1}\simeq {H}_{q}\left({C}_{p,•}\right)$;

• ${E}_{p,q}^{2}\simeq {H}_{p}\left({H}_{q}^{\mathrm{vert}}\left(C\right)\right)$.

Moreover, if ${C}_{•,•}$ is concentrated in the first quadrant ($0\le p,q$), then the spectral sequence converges to the chain homology of the total complex:

${E}_{p,q}^{\infty }\simeq {G}_{p}{H}_{p+q}\left(\mathrm{Tot}\left(C{\right)}_{•}\right)\phantom{\rule{thinmathspace}{0ex}}.$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{array}{rl}{E}_{p,q}^{0}& ≔{G}_{p}\mathrm{Tot}\left(C{\right)}_{p+q}\\ & ≔{F}_{p}\mathrm{Tot}\left(C{\right)}_{p+q}/{F}_{p-1}\mathrm{Tot}\left(C{\right)}_{p+q}\\ & ≔\frac{\underset{\genfrac{}{}{0}{}{{n}_{1}+{n}_{2}=p+q}{{n}_{1}\le p}}{\oplus }{C}_{{n}_{1},{n}_{2}}}{\underset{\genfrac{}{}{0}{}{{n}_{1}+{n}_{2}=p+q}{{n}_{1}\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:

$\begin{array}{rl}{E}_{p,q}^{1}& \simeq {H}_{p+q}\left({G}_{p}\mathrm{Tot}\left(C{\right)}_{•}\right)\\ & \simeq {H}_{p+q}\left({C}_{p,•}\right)\\ & \simeq {H}_{q}\left({C}_{p,•}\right)\end{array}\phantom{\rule{thinmathspace}{0ex}}.$\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 $\left[c\right]\in {E}_{p,q}^{1}$ are given by $c\in {C}_{p,q}$ such that ${\partial }^{\mathrm{vert}}c=0$. It follows that ${\partial }^{1}:{E}_{p,q}^{1}\to {E}_{p-1,q}^{1}$, which by the definition of a total complex acts as ${\partial }^{\mathrm{hor}}±{\partial }^{\mathrm{vert}}$, acts on these representatives just as ${\partial }^{\mathrm{hor}}$ and this gives the second page

${E}_{p,q}^{2}\simeq \mathrm{ker}\left({\partial }_{p-1,q}^{1}\right)/\mathrm{im}\left({\partial }_{p,q}^{1}\right)\simeq {H}_{p}\left({H}_{q}^{\mathrm{vert}}\left({C}_{•,•}\right)\right)\phantom{\rule{thinmathspace}{0ex}}.$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 ${C}_{•,•}$ concentrated in $0\le p,q$ the corresponding filtered chain complex ${F}_{p}\mathrm{Tot}\left(C{\right)}_{•}$ 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 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

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

Revised on October 29, 2012 17:04:47 by Urs Schreiber (131.174.188.167)