nLab Grothendieck spectral sequence

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

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)≃R p+q(G∘F)(A). E^{p,q}_\infty(A) \simeq 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, consider 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 GG-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

Last revised on April 22, 2020 at 13:17:00. See the history of this page for a list of all contributions to it.