Grothendieck spectral sequence

Short description: Spectral sequence

In mathematics, in the field of homological algebra, the Grothendieck spectral sequence, introduced by Alexander Grothendieck in his Tôhoku paper, is a spectral sequence that computes the derived functors of the composition of two functors G∘F, from knowledge of the derived functors of F and G. Many spectral sequences in algebraic geometry are instances of the Grothendieck spectral sequence, for example the Leray spectral sequence.

## Contents

* 1 Statement
* 1.1 Five term exact sequence
* 2 Examples
* 2.1 The Leray spectral sequence
* 2.2 Local-to-global Ext spectral sequence
* 3 Derivation
* 4 Notes
* 5 References
* 5.1 Computational Examples

## Statement

If F:𝒜->ℬ and G:ℬ->𝒞 are two additive and left exact functors between abelian categories such that both 𝒜 and ℬ have enough injectives and F takes injective objects to G-acyclic objects, then for each object A of 𝒜 there is a spectral sequence:

E2pq=(RpG∘RqF)(A)⟹Rp+q(G∘F)(A),

where RpG denotes the p-th right-derived functor of G, etc., and where the arrow '⟹' means convergence of spectral sequences.

### Five term exact sequence

The exact sequence of low degrees reads

0->R1G(FA)->R1(GF)(A)->G(R1F(A))->R2G(FA)->R2(GF)(A).

## Examples

### The Leray spectral sequence

Main article: Leray spectral sequence

If X and Y are topological spaces, let 𝒜=𝐀𝐛(X) and ℬ=𝐀𝐛(Y) be the category of sheaves of abelian groups on X and Y, respectively.

For a continuous map f:X->Y there is the (left-exact) direct image functor f*:𝐀𝐛(X)->𝐀𝐛(Y). We also have the global section functors

ΓX:𝐀𝐛(X)->𝐀𝐛 and ΓY:𝐀𝐛(Y)->𝐀𝐛.

Then since ΓY∘f*=ΓX and the functors f* and ΓY satisfy the hypotheses (since the direct image functor has an exact left adjoint f−1, pushforwards of injectives are injective and in particular acyclic for the global section functor), the sequence in this case becomes:

Hp(Y,Rqf*ℱ)⟹Hp+q(X,ℱ)

for a sheaf ℱ of abelian groups on X.

### Local-to-global Ext spectral sequence

There is a spectral sequence relating the global Ext and the sheaf Ext: let F, G be sheaves of modules over a ringed space (X,𝒪); e.g., a scheme. Then

E2p,q=Hp(X;ℰxt𝒪q(F,G))⇒Ext𝒪p+q(F,G).[1]

This is an instance of the Grothendieck spectral sequence: indeed,

RpΓ(X,−)=Hp(X,−), Rqℋom𝒪(F,−)=ℰxt𝒪q(F,−) and RnΓ(X,ℋom𝒪(F,−))=Ext𝒪n(F,−).

Moreover, ℋom𝒪(F,−) sends injective 𝒪-modules to flasque sheaves,[2] which are Γ(X,−)-acyclic. Hence, the hypothesis is satisfied.

## Derivation

We shall use the following lemma:

Lemma -- If K is an injective complex in an abelian category C such that the kernels of the differentials are injective objects, then for each n,

Hn(K∙)

is an injective object and for any left-exact additive functor G on C,

Hn(G(K∙))=G(Hn(K∙)).

Proof: Let Zn,Bn+1 be the kernel and the image of d:Kn->Kn+1. We have

0->Zn->Kn->dBn+1->0,

which splits. This implies each Bn+1 is injective. Next we look at

0->Bn->Zn->Hn(K∙)->0.

It splits, which implies the first part of the lemma, as well as the exactness of

0->G(Bn)->G(Zn)->G(Hn(K∙))->0.

Similarly we have (using the earlier splitting):

0->G(Zn)->G(Kn)->G(d)G(Bn+1)->0.

The second part now follows. ◻

We now construct a spectral sequence. Let A0->A1->⋯ be an injective resolution of A. Writing ϕp for F(Ap)->F(Ap+1), we have:

0->ker⁡ϕp->F(Ap)->ϕpim⁡ϕp->0.

Take injective resolutions J0->J1->⋯ and K0->K1->⋯ of the first and the third nonzero terms. By the horseshoe lemma, their direct sum Ip,∙=J⊕K is an injective resolution of F(Ap). Hence, we found an injective resolution of the complex:

0->F(A∙)->I∙,0->I∙,1->⋯.

such that each row I0,q->I1,q->⋯ satisfies the hypothesis of the lemma (cf. the Cartan–Eilenberg resolution.)

Now, the double complex E0p,q=G(Ip,q) gives rise to two spectral sequences, horizontal and vertical, which we are now going to examine. On the one hand, by definition,

′′E1p,q=Hq(G(Ip,∙))=RqG(F(Ap)),

which is always zero unless q = 0 since F(Ap) is G-acyclic by hypothesis. Hence, ′′E2n=Rn(G∘F)(A) and ′′E2=′′E∞. On the other hand, by the definition and the lemma,

′E1p,q=Hq(G(I∙,p))=G(Hq(I∙,p)).

Since Hq(I∙,0)->Hq(I∙,1)->⋯ is an injective resolution of Hq(F(A∙))=RqF(A) (it is a resolution since its cohomology is trivial),

′E2p,q=RpG(RqF(A)).

Since ′Er and ′′Er have the same limiting term, the proof is complete. ◻

## Notes

1. ↑ Godement 1973, Ch. II, Theorem 7.3.3.
2. ↑ Godement 1973, Ch. II, Lemma 7.3.2.

## References

* Godement, Roger (1973), Topologie algébrique et théorie des faisceaux, Paris: Hermann
* Weibel, Charles A. (1994). An introduction to homological algebra. Cambridge Studies in Advanced Mathematics. 38. Cambridge University Press. ISBN 978-0-521-55987-4. OCLC 36131259.

### Computational Examples

* Sharpe, Eric (2003). Lectures on D-branes and Sheaves (pages 18–19), arXiv:hep-th/0307245

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