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Statement

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

E 2 p q = ( R p G ∘ R q F ) ( A ) ⟹ R p + q ( G ∘ F ) ( A ) , {\displaystyle E_{2}^{pq}=({\rm {R}}^{p}G\circ {\rm {R}}^{q}F)(A)\Longrightarrow {\rm {R}}^{p+q}(G\circ F)(A),}

where R p G {\displaystyle {\rm {R}}^{p}G} denotes the p-th right-derived functor of G {\displaystyle G} , etc., and where the arrow ' ⟹ {\displaystyle \Longrightarrow } ' means convergence of spectral sequences.

Five term exact sequence

The exact sequence of low degrees reads

0 → R 1 G ( F A ) → R 1 ( G F ) ( A ) → G ( R 1 F ( A ) ) → R 2 G ( F A ) → R 2 ( G F ) ( A ) . {\displaystyle 0\to {\rm {R}}^{1}G(FA)\to {\rm {R}}^{1}(GF)(A)\to G({\rm {R}}^{1}F(A))\to {\rm {R}}^{2}G(FA)\to {\rm {R}}^{2}(GF)(A).}

Examples

The Leray spectral sequence

Main article: Leray spectral sequence

If X {\textstyle X} and Y {\textstyle Y} are topological spaces, let A = A b ( X ) {\textstyle {\mathcal {A}}=\mathbf {Ab} (X)} and B = A b ( Y ) {\textstyle {\mathcal {B}}=\mathbf {Ab} (Y)} be the category of sheaves of abelian groups on X {\textstyle X} and Y {\textstyle Y} , respectively.

For a continuous map f : X → Y {\displaystyle f\colon X\to Y} there is the (left-exact) direct image functor f ∗ : A b ( X ) → A b ( Y ) {\displaystyle f_{*}\colon \mathbf {Ab} (X)\to \mathbf {Ab} (Y)} . We also have the global section functors

Γ X : A b ( X ) → A b {\displaystyle \Gamma _{X}\colon \mathbf {Ab} (X)\to \mathbf {Ab} } and Γ Y : A b ( Y ) → A b . {\displaystyle \Gamma _{Y}\colon \mathbf {Ab} (Y)\to \mathbf {Ab} .}

Then since Γ Y ∘ f ∗ = Γ X {\displaystyle \Gamma _{Y}\circ f_{*}=\Gamma _{X}} and the functors f ∗ {\displaystyle f_{*}} and Γ Y {\displaystyle \Gamma _{Y}} satisfy the hypotheses (since the direct image functor has an exact left adjoint f − 1 {\displaystyle 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 ) {\displaystyle H^{p}(Y,{\rm {R}}^{q}f_{*}{\mathcal {F}})\implies H^{p+q}(X,{\mathcal {F}})}

for a sheaf F {\displaystyle {\mathcal {F}}} of abelian groups on X {\displaystyle 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 ) {\displaystyle (X,{\mathcal {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 ) . {\displaystyle E_{2}^{p,q}=\operatorname {H} ^{p}(X;{\mathcal {E}}xt_{\mathcal {O}}^{q}(F,G))\Rightarrow \operatorname {Ext} _{\mathcal {O}}^{p+q}(F,G).} ¹

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

R p Γ ( X , − ) = H p ⁡ ( X , − ) {\displaystyle R^{p}\Gamma (X,-)=\operatorname {H} ^{p}(X,-)} , R q H o m O ( F , − ) = E x t O q ( F , − ) {\displaystyle R^{q}{\mathcal {H}}om_{\mathcal {O}}(F,-)={\mathcal {E}}xt_{\mathcal {O}}^{q}(F,-)} and R n Γ ( X , H o m O ( F , − ) ) = Ext O n ⁡ ( F , − ) {\displaystyle R^{n}\Gamma (X,{\mathcal {H}}om_{\mathcal {O}}(F,-))=\operatorname {Ext} _{\mathcal {O}}^{n}(F,-)} .

Moreover, H o m O ( F , − ) {\displaystyle {\mathcal {H}}om_{\mathcal {O}}(F,-)} sends injective O {\displaystyle {\mathcal {O}}} -modules to flasque sheaves,² which are Γ ( X , − ) {\displaystyle \Gamma (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,

H n ( K ∙ ) {\displaystyle H^{n}(K^{\bullet })}

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

H n ( G ( K ∙ ) ) = G ( H n ( K ∙ ) ) . {\displaystyle H^{n}(G(K^{\bullet }))=G(H^{n}(K^{\bullet })).}

Proof: Let Z n , B n + 1 {\displaystyle Z^{n},B^{n+1}} be the kernel and the image of d : K n → K n + 1 {\displaystyle d:K^{n}\to K^{n+1}} . We have

0 → Z n → K n → d B n + 1 → 0 , {\displaystyle 0\to Z^{n}\to K^{n}{\overset {d}{\to }}B^{n+1}\to 0,}

which splits. This implies each B n + 1 {\displaystyle B^{n+1}} is injective. Next we look at

0 → B n → Z n → H n ( K ∙ ) → 0. {\displaystyle 0\to B^{n}\to Z^{n}\to H^{n}(K^{\bullet })\to 0.}

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

0 → G ( B n ) → G ( Z n ) → G ( H n ( K ∙ ) ) → 0. {\displaystyle 0\to G(B^{n})\to G(Z^{n})\to G(H^{n}(K^{\bullet }))\to 0.}

Similarly we have (using the earlier splitting):

0 → G ( Z n ) → G ( K n ) → G ( d ) G ( B n + 1 ) → 0. {\displaystyle 0\to G(Z^{n})\to G(K^{n}){\overset {G(d)}{\to }}G(B^{n+1})\to 0.}

The second part now follows. ◻ {\displaystyle \square }

We now construct a spectral sequence. Let A 0 → A 1 → ⋯ {\displaystyle A^{0}\to A^{1}\to \cdots } be an injective resolution of A. Writing ϕ p {\displaystyle \phi ^{p}} for F ( A p ) → F ( A p + 1 ) {\displaystyle F(A^{p})\to F(A^{p+1})} , we have:

0 → ker ⁡ ϕ p → F ( A p ) → ϕ p im ⁡ ϕ p → 0. {\displaystyle 0\to \operatorname {ker} \phi ^{p}\to F(A^{p}){\overset {\phi ^{p}}{\to }}\operatorname {im} \phi ^{p}\to 0.}

Take injective resolutions J 0 → J 1 → ⋯ {\displaystyle J^{0}\to J^{1}\to \cdots } and K 0 → K 1 → ⋯ {\displaystyle K^{0}\to K^{1}\to \cdots } of the first and the third nonzero terms. By the horseshoe lemma, their direct sum I p , ∙ = J ⊕ K {\displaystyle I^{p,\bullet }=J\oplus K} is an injective resolution of F ( A p ) {\displaystyle F(A^{p})} . Hence, we found an injective resolution of the complex:

0 → F ( A ∙ ) → I ∙ , 0 → I ∙ , 1 → ⋯ . {\displaystyle 0\to F(A^{\bullet })\to I^{\bullet ,0}\to I^{\bullet ,1}\to \cdots .}

such that each row I 0 , q → I 1 , q → ⋯ {\displaystyle I^{0,q}\to I^{1,q}\to \cdots } satisfies the hypothesis of the lemma (cf. the Cartan–Eilenberg resolution.)

Now, the double complex E 0 p , q = G ( I p , q ) {\displaystyle 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 ) ) {\displaystyle {}^{\prime \prime }E_{1}^{p,q}=H^{q}(G(I^{p,\bullet }))=R^{q}G(F(A^{p}))} ,

which is always zero unless q = 0 since F ( A p ) {\displaystyle F(A^{p})} is G-acyclic by hypothesis. Hence, ′ ′ E 2 n = R n ( G ∘ F ) ( A ) {\displaystyle {}^{\prime \prime }E_{2}^{n}=R^{n}(G\circ F)(A)} and ′ ′ E 2 = ′ ′ E ∞ {\displaystyle {}^{\prime \prime }E_{2}={}^{\prime \prime }E_{\infty }} . On the other hand, by the definition and the lemma,

′ E 1 p , q = H q ( G ( I ∙ , p ) ) = G ( H q ( I ∙ , p ) ) . {\displaystyle {}^{\prime }E_{1}^{p,q}=H^{q}(G(I^{\bullet ,p}))=G(H^{q}(I^{\bullet ,p})).}

Since H q ( I ∙ , 0 ) → H q ( I ∙ , 1 ) → ⋯ {\displaystyle H^{q}(I^{\bullet ,0})\to H^{q}(I^{\bullet ,1})\to \cdots } is an injective resolution of H q ( F ( A ∙ ) ) = R q F ( A ) {\displaystyle H^{q}(F(A^{\bullet }))=R^{q}F(A)} (it is a resolution since its cohomology is trivial),

′ E 2 p , q = R p G ( R q F ( A ) ) . {\displaystyle {}^{\prime }E_{2}^{p,q}=R^{p}G(R^{q}F(A)).}

Since ′ E r {\displaystyle {}^{\prime }E_{r}} and ′ ′ E r {\displaystyle {}^{\prime \prime }E_{r}} have the same limiting term, the proof is complete. ◻ {\displaystyle \square }

Notes

• Godement, Roger (1973), Topologie algébrique et théorie des faisceaux, Paris: Hermann, MR 0345092
• Weibel, Charles A. (1994). An introduction to homological algebra. Cambridge Studies in Advanced Mathematics. Vol. 38. Cambridge University Press. ISBN 978-0-521-55987-4. MR 1269324. OCLC 36131259.

Computational Examples

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

This article incorporates material from Grothendieck spectral sequence on PlanetMath, which is licensed under the Creative Commons Attribution/Share-Alike License.

References

1. Godement 1973, Ch. II, Theorem 7.3.3. - Godement, Roger (1973), Topologie algébrique et théorie des faisceaux, Paris: Hermann, MR 0345092 https://mathscinet.ams.org/mathscinet-getitem?mr=0345092 ↩

2. Godement 1973, Ch. II, Lemma 7.3.2. - Godement, Roger (1973), Topologie algébrique et théorie des faisceaux, Paris: Hermann, MR 0345092 https://mathscinet.ams.org/mathscinet-getitem?mr=0345092 ↩