In mathematics, in the field of homological algebra, the Grothendieck spectral sequence, introduced in 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.
If F : A → B and G : B → C are two additive and left exact functors between abelian categories such that F takes F-acyclic objects (e.g., injective objects) to G -acyclic objects and if B has enough injectives, then there is a spectral sequence for each object A of A that admits an F-acyclic resolution:
E 2 p q = ( R p G ∘ R q F ) ( A ) ⟹ R p + q ( G ∘ F ) ( A ) . Many spectral sequences in algebraic geometry are instances of the Grothendieck spectral sequence, for example the Leray spectral sequence.
The exact sequence of low degrees reads
0 →
R1G(
FA) →
R1(
GF)(
A) →
G(
R1F(
A)) →
R2G(
FA) →
R2(
GF)(
A).
If X and Y are topological spaces, let
A = A b ( X ) and
B = A b ( Y ) be the category of sheaves of abelian groups on
X and
Y, respectively and
C = A b be the category of abelian groups.
For a continuous map
f : X → Y there is the (left-exact) direct image functor
f ∗ : A b ( X ) → A b ( Y ) .
We also have the global section functors
Γ X : A b ( X ) → A b ,
and
Γ Y : A b ( Y ) → A b . Then since
Γ Y ∘ f ∗ = Γ X and the functors f ∗ and Γ Y 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:
H p ( Y , R q f ∗ F ) ⟹ H p + q ( X , F ) for a sheaf F of abelian groups on X , and this is exactly the Leray 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
E 2 p , q = H p ( X ; E x t O q ( F , G ) ) ⇒ Ext O p + q ( F , G ) . This is an instance of the Grothendieck spectral sequence: indeed,
R p Γ ( X , − ) = H p ( X , − ) ,
R q H o m O ( F , − ) = E x t O q ( F , − ) and
R n Γ ( X , H o m O ( F , − ) ) = Ext O n ( F , − ) .
Moreover, H o m O ( F , − ) sends injective O -modules to flasque sheaves, which are Γ ( X , − ) -acyclic. Hence, the hypothesis is satisfied.
We shall use the following lemma:
Proof: Let Z n , B n + 1 be the kernel and the image of d : K n → K n + 1 . We have
0 → Z n → K n → d B n + 1 → 0 ,
which splits and implies B n + 1 is injective and the first part of the lemma. Next we look at
0 → B n → Z n → H n ( K ∙ ) → 0. It splits. Thus,
0 → G ( B n ) → G ( Z n ) → G ( H n ( K ∙ ) ) → 0. Similarly we have (using the early splitting):
0 → G ( Z n ) → G ( K n ) → G ( d ) G ( B n + 1 ) → 0 .
The second part now follows. ◻
We now construct a spectral sequence. Let A 0 → A 1 → ⋯ be an F-acyclic resolution of A. Writing ϕ p for F ( A p ) → F ( A p + 1 ) , we have:
0 → ker ϕ p → F ( A p ) → ϕ p im ϕ p → 0. Take injective resolutions J 0 → J 1 → ⋯ and K 0 → K 1 → ⋯ of the first and the third nonzero terms. By the horseshoe lemma, their direct sum I p , ∙ = J ⊕ K is an injective resolution of F ( A p ) . Hence, we found an injective resolution of the complex:
0 → F ( A ∙ ) → I ∙ , 0 → I ∙ , 1 → ⋯ . such that each row I 0 , q → I 1 , q → ⋯ satisfies the hypothesis of the lemma (cf. the Cartan–Eilenberg resolution.)
Now, the double complex E 0 p , q = G ( I p , q ) gives rise to two spectral sequences, horizontal and vertical, which we are now going to examine. On the one hand, by definition,
′ ′ E 1 p , q = H q ( G ( I p , ∙ ) ) = R q G ( F ( A p ) ) ,
which is always zero unless q = 0 since F ( A p ) is G-acyclic by hypothesis. Hence, ′ ′ E 2 n = R n ( G ∘ F ) ( A ) and ′ ′ E 2 = ′ ′ E ∞ . On the other hand, by the definition and the lemma,
′ E 1 p , q = H q ( G ( I ∙ , p ) ) = G ( H q ( I ∙ , p ) ) . Since H q ( I ∙ , 0 ) → H q ( I ∙ , 1 ) → ⋯ is an injective resolution of H q ( F ( A ∙ ) ) = R q F ( A ) (it is a resolution since its cohomology is trivial),
′ E 2 p , q = R p G ( R q F ( A ) ) . Since ′ E r and ′ ′ E r have the same limiting term, the proof is complete. ◻
