spectral sequence of a double complex in nLab

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Homological algebra

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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 1≤pC 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 2≤pC 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} \,.

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,q∈ℤp,q \in \mathbb{Z} given by

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

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

• E p,q 2≃H 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 (0≤p,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 p−1Tot(C) p+q ≔⊕n 1+n 2=p+qn 1≤pC n 1,n 2⊕n 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 c∈C p,qc \in C_{p,q} such that ∂ vertc=0\partial^{vert} c = 0. It follows that ∂ 1:E p,q 1→E p−1,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 2≃ker(∂ p−1,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 0≤p,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.