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Grothendieck spectral sequence

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Idea

A Grothendieck spectral sequence is a spectral sequence that computes the cochain cohomology of the composite of two derived functors on categories of chain complexes.

Statement

Let π’œ,ℬ,π’ž\mathcal{A},\mathcal{B},\mathcal{C} be abelian categories and let F:π’œβ†’β„¬F \colon \mathcal{A}\to \mathcal{B} and G:β„¬β†’π’žG \colon \mathcal{B}\to \mathcal{C} be left exact additive functors. Assume that π’œ,ℬ\mathcal{A}, \mathcal{B} have enough injectives.

Theorem (Tohoku)

Write R FβŠ‚ObAR_F\subset \mathrm{Ob} A and R GβŠ‚ObBR_G\subset\mathrm{Ob} B for the classes of objects adapted to FF and GG respectively, and let furthermore F(R A)βŠ‚R BF(R_A)\subset R_B. Then the derived functors RF:D +(A)β†’D +(B)R F:D^+(A)\to D^+(B), RG:D +(B)β†’D +(C)R G:D^+(B)\to D^+(C) and R(G∘F):D +(A)β†’D +(C)R(G\circ F):D^+(A)\to D^+(C) are defined and the natural morphism R(G∘F)β†’RG∘RFR(G\circ F)\to R G\circ R F is an isomorphism.

Theorem (Tohoku)

In the above situation, assume that for every injective object Iβˆˆπ’œI \in \mathcal{A} the object F(I)βˆˆβ„¬F(I) \in \mathcal{B} is a GG-acyclic object.

Then for every object Aβˆˆπ’œA \in \mathcal{A} there is a spectral sequence {E p,q r(A)} r,p,q\{E^r_{p,q}(A)\}_{r,p,q} called the Grothendieck spectral sequence whose E 2E_2-page is the composite

E 2 p,q(A)=R pG∘R qF(A) E^{p,q}_2(A) = R^p G \circ R^q F (A)

of the right derived functors of FF and GG in degrees qq and pp, respectively and which is converging to to the derived functors R n(G∘F)R^n(G\circ F) of the composite of FF and GG:

E ∞ p,q(A)≃G pR p+q(G∘F)(A). E^{p,q}_\infty(A) \simeq G^p R^{p+q}(G \circ F)(A) \,.

Moreover, this is natural in Aβˆˆπ’œA \in \mathcal{A}.

Proof

By assumption of enough injectives, we may find an injective resolution

A→≃ qiC β€’ A \stackrel{\simeq_{qi}}{\to} C^\bullet

of AA. Next, by the discussion at injective resolution – Existence and construction we may find a fully injective resolution of the chain complex F(C β€’)F(C^\bullet):

0β†’F(C β€’)β†’I β€’,β€’, 0 \to F(C^\bullet) \to I^{\bullet, \bullet} \,,

where hence I β€’,β€’I^{\bullet, \bullet} is a double complex of injective objects such that for each nβˆˆβ„•n \in \mathbb{N} the component 0β†’F(C n)β†’I n,β€’0 \to F(C^n) \to I^{n,\bullet} is an ordinary injective resolution of F(C n)βˆˆβ„¬F(C^n) \in \mathcal{B}.

Thus we have the corresponding double complex G(I β€’,β€’)G(I^{\bullet,\bullet}) in π’ž\mathcal{C}. The claim is that the Grothendieck spectral sequence is the spectral sequence of a double complex for G(I β€’,β€’)G(I^{\bullet, \bullet}) equipped with the vertical-degree filtration { vertE p,q r(A)}\{{}^{vert}E^r_{p,q}(A)\}:

vertE p,q 2(A)≃R pG(R qF(A)). {}^{vert} E^2_{p,q}(A) \simeq R^p G (R^q F(A)) \,.

To see this, notice that by the assumption that I β€’,β€’I^{\bullet,\bullet} is a fully injective projective resolution, the short exact sequences

0β†’B q,p(I)β†’Z q,p(I)β†’H q,p(I)β†’0 0 \to B^{q,p}(I) \to Z^{q,p}(I) \to H^{q,p}(I) \to 0

are split (by the discussion there) and hence so is their image under any functor and hence in particular under GG. Accordingly we have

vertE 1 p,q ≃H q(G(I β€’,p)) ≃(G(Z q,p))/(G(B q,p)) ≃GH q,p \begin{aligned} {}^{vert}E^{p,q}_1 & \simeq H^q(G(I^{\bullet,p})) \\ & \simeq (G(Z^{q,p})) / (G(B^{q,p})) \\ & \simeq G H^{q,p} \end{aligned}

(the first two equivalences by general properties of the filtration spectral sequence, the last by the above splitness). Hence it follows that

vertE 2 p,q ≃H p(G(H q,β€’)) ≃R pG(R qF(A)), \begin{aligned} {}^{vert}E^{p,q}_2 & \simeq H^p(G(H^{q,\bullet})) \\ & \simeq R^p G (R^q F (A)) \end{aligned} \,,

where in the last step we used that H q,β€’H^{q,\bullet} is be construction an injective resolution of H q(F(C β€’))≃R qF(A)H^q(F(C^\bullet)) \simeq R^q F(A) (using the GG-acyclicity of F(C β€’)F(C^\bullet)).

This establishes the spectral sequence and its second page as claimed. It remains to determine its convergence.

To that end, conside dually, the spectral sequence { horE r p,q}\{{}^{hor}E^{p,q}_r\} coming from the horizontal filtration on the double complex G(I β€’,β€’)G(I^{\bullet, \bullet}). By the general properties of spectral sequence of a double complex this converges to the same value as the previous one. But for this latter spectral sequence we find

horE 1 p,q ≃H q(GI p,β€’) ≃R qG(F(C p)), \begin{aligned} {}^{hor}E^{p,q}_1 & \simeq H^q(G I^{p,\bullet}) \\ & \simeq R^q G(F(C^p)) \end{aligned} \,,

the first equivalence by the general properties of filtration spectral sequences, the second then by the definition of right derived functors. But by assumption F(C p)F(C^p) is FF-acyclic and hence all these derived functors vanish in positive degree, so that

horE 1 p,q≃{G(F(C p)) ifq=0 0 otherwise. {}^{hor}E^{p,q}_1 \simeq \left\{ \array{ G(F(C^p)) & if\; q = 0 \\ 0 & otherwise } \right. \,.

Next, the E 2E_2-page then contains just horizontal homology of G(F(C β€’))G(F(C^\bullet)) and this is by definition now the derived functor of the composite of FF with GG:

horE 2 p,q≃{R p(G∘F) ifq=0 0 otherwise. {}^{hor}E^{p,q}_2 \simeq \left\{ \array{ R^p(G \circ F) & if \; q = 0 \\ 0 & otherwise } \right. \,.

Since this is concentrated in the (q=0)(q = 0)-row the spectral sequence of the horizontal filtration collapses here and hence

H n(Tot(G(I β€’,β€’))) ≃G nH n+0(Tot(G(I β€’,β€’))) ≃E ∞ n,0 \begin{aligned} H^n(Tot(G(I^{\bullet,\bullet}))) &\simeq G^n H^{n+0}(Tot(G(I^{\bullet,\bullet}))) \\ & \simeq E^{n,0}_\infty \end{aligned}

So in conclusion we have

R pG(R qF(A)) ≃ vertE 2 p,q β‡’ vertE ∞ p,q ≃G vert pH p+q(Tot(G(I β€’,β€’))) ≃H p+q(Tot(G(I β€’,β€’))) ≃G hor p+qH p+q(Tot(G(I β€’,β€’))) ≃ horE ∞ p+q,0(A) ≃R p+q(G∘F)(A) \begin{aligned} R^p G(R^q F(A)) & \simeq {}^{vert}E^{p,q}_2 \\ & \Rightarrow {}^{vert} E^{p,q}_\infty \\ & \simeq G^p_{vert} H^{p+q}(Tot(G(I^{\bullet, \bullet}))) \\ & \simeq H^{p+q}(Tot(G(I^{\bullet, \bullet}))) \\ & \simeq G^{p+q}_{hor} H^{p+q}(Tot(G(I^{\bullet, \bullet}))) \\ & \simeq {}^{hor} E^{p+q,0}_\infty(A) \\ & \simeq R^{p+q}(G \circ F)(A) \end{aligned}

Examples

Many other classes of spectral sequences are special cases of the Grothendieck spectral sequence, for instance the

References

Leture notes include

Revised on November 29, 2014 19:41:03 by Ingo Blechschmidt (137.250.162.16)