nLab spectral sequence of a double complex

Contents

Context

Homological algebra

homological algebra

(also nonabelian homological algebra)

Introduction

Context

Basic definitions

Stable homotopy theory notions

Constructions

Lemmas

diagram chasing

Schanuel's lemma

Homology theories

Theorems

Contents

Idea

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

E p,q 2=H p hor(H q vert(C)) 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 ,)Tot(C_{\bullet, \bullet}).

This is the special case of the spectral sequence of a filtered complex with Tot(C) Tot(C)_\bullet filtered by row-degree (or dually, by column-degree).

Definition

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

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

and the differential acts on elements cC p,qc \in C_{p,q} as

Tot:c horc+(1) p verc. \partial^{Tot} \colon c \mapsto \partial^{hor} c + (-1)^{p} \partial^{ver} c \,.
Definition

The horizontal filtration on TotCTot C is the filtration F TotCF_\bullet Tot C given in degree nn by the direct sum expression

F p hor(TotC) n n 1+n 2=nn 1pC n 1,n 2. 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 p vert(TotC) n n 1+n 2=nn 2pC n 1,n 2. 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 ,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 p,q r} r,p,q\{E^r_{p,q}\}_{r,p,q} be the spectral sequence of a double complex C ,C_{\bullet, \bullet}, according to def. , with respect to the horizontal filtration. Then the first few pages are for all p,qp,q \in \mathbb{Z} given by

  • E p,q 0C p,qE^0_{p,q} \simeq C_{p,q};

  • E p,q 1H q(C p,)E^1_{p,q} \simeq H_q(C_{p, \bullet});

  • E p,q 2H p(H q vert(C))E^2_{p,q} \simeq H_p(H^{vert}_q(C)).

Moreover, if C ,C_{\bullet, \bullet} is concentrated in the first quadrant (0p,q0 \leq p,q), then the spectral sequence converges to the chain homology of the total complex:

E p,q G pH p+q(Tot(C) ). 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

E p,q 0 G pTot(C) p+q F pTot(C) p+q/F p1Tot(C) p+q n 1+n 2=p+qn 1pC n 1,n 2n 1+n 2=p+qn 1<pC n 1,n 2 C p,q. \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:

E p,q 1 H p+q(G pTot(C) ) H p+q(C p,) H q(C p,). \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]E p,q 1[c] \in E^1_{p,q} are given by cC p,qc \in C_{p,q} such that vertc=0\partial^{vert} c = 0. It follows that 1:E p,q 1E p1,q 1\partial^1 \colon E^1_{p,q} \to E^1_{p-1, q}, which by the definition of a total complex acts as hor± vert\partial^{hor} \pm \partial^{vert}, acts on these representatives just as hor\partial^{hor} and this gives the second page

E p,q 2ker( p1,q 1)/im( p,q 1)H p(H q vert(C ,)). 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 ,C_{\bullet, \bullet} concentrated in 0p,q0 \leq p,q the corresponding filtered chain complex F pTot(C) 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

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.

Last revised on February 21, 2018 at 16:53:59. See the history of this page for a list of all contributions to it.