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spectral sequence of a double complex

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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) nCn 1+n 2=nn 1p n 1,n 2. 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 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. 1.

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. 2, 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 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)