Grothendieck spectral sequence

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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 ${\displaystyle G\circ F}${\displaystyle G\circ F}, from knowledge of the derived functors of ${\displaystyle F}${\displaystyle F} and ${\displaystyle G}${\displaystyle G}. Many spectral sequences in algebraic geometry are instances of the Grothendieck spectral sequence, for example the Leray spectral sequence.

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

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

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

The exact sequence of low degrees reads

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

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

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

${\displaystyle \Gamma _{X}\colon \mathbf {Ab} (X)\to \mathbf {Ab} }${\displaystyle \Gamma _{X}\colon \mathbf {Ab} (X)\to \mathbf {Ab} } and ${\displaystyle \Gamma _{Y}\colon \mathbf {Ab} (Y)\to \mathbf {Ab} .}${\displaystyle \Gamma _{Y}\colon \mathbf {Ab} (Y)\to \mathbf {Ab} .}

Then since ${\displaystyle \Gamma _{Y}\circ f_{*}=\Gamma _{X}}${\displaystyle \Gamma _{Y}\circ f_{*}=\Gamma _{X}} and the functors ${\displaystyle f_{*}}${\displaystyle f_{*}} and ${\displaystyle \Gamma _{Y}}${\displaystyle \Gamma _{Y}} satisfy the hypotheses (since the direct image functor has an exact left adjoint ${\displaystyle f^{-1}}${\displaystyle f^{-1}}, pushforwards of injectives are injective and in particular acyclic for the global section functor), the sequence in this case becomes:

${\displaystyle H^{p}(Y,{\rm {R}}^{q}f_{*}{\mathcal {F}})\implies H^{p+q}(X,{\mathcal {F}})}${\displaystyle H^{p}(Y,{\rm {R}}^{q}f_{*}{\mathcal {F}})\implies H^{p+q}(X,{\mathcal {F}})}

for a sheaf ${\displaystyle {\mathcal {F}}}${\displaystyle {\mathcal {F}}} of abelian groups on ${\displaystyle X}${\displaystyle X}.

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

${\displaystyle E_{2}^{p,q}=\operatorname {H} ^{p}(X;{\mathcal {E}}xt_{\mathcal {O}}^{q}(F,G))\Rightarrow \operatorname {Ext} _{\mathcal {O}}^{p+q}(F,G).}${\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,

${\displaystyle R^{p}\Gamma (X,-)=\operatorname {H} ^{p}(X,-)}${\displaystyle R^{p}\Gamma (X,-)=\operatorname {H} ^{p}(X,-)}, ${\displaystyle R^{q}{\mathcal {H}}om_{\mathcal {O}}(F,-)={\mathcal {E}}xt_{\mathcal {O}}^{q}(F,-)}${\displaystyle R^{q}{\mathcal {H}}om_{\mathcal {O}}(F,-)={\mathcal {E}}xt_{\mathcal {O}}^{q}(F,-)} and ${\displaystyle R^{n}\Gamma (X,{\mathcal {H}}om_{\mathcal {O}}(F,-))=\operatorname {Ext} _{\mathcal {O}}^{n}(F,-)}${\displaystyle R^{n}\Gamma (X,{\mathcal {H}}om_{\mathcal {O}}(F,-))=\operatorname {Ext} _{\mathcal {O}}^{n}(F,-)}.

Moreover, ${\displaystyle {\mathcal {H}}om_{\mathcal {O}}(F,-)}${\displaystyle {\mathcal {H}}om_{\mathcal {O}}(F,-)} sends injective ${\displaystyle {\mathcal {O}}}${\displaystyle {\mathcal {O}}}-modules to flasque sheaves,^[2] which are ${\displaystyle \Gamma (X,-)}${\displaystyle \Gamma (X,-)}-acyclic. Hence, the hypothesis is satisfied.

We shall use the following lemma:

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

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

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

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

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

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

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

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

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

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

such that each row ${\displaystyle I^{0,q}\to I^{1,q}\to \cdots }${\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 ${\displaystyle E_{0}^{p,q}=G(I^{p,q})}${\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,

${\displaystyle {}^{\prime \prime }E_{1}^{p,q}=H^{q}(G(I^{p,\bullet }))=R^{q}G(F(A^{p}))}${\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 ${\displaystyle F(A^{p})}${\displaystyle F(A^{p})} is G-acyclic by hypothesis. Hence, ${\displaystyle {}^{\prime \prime }E_{2}^{n}=R^{n}(G\circ F)(A)}${\displaystyle {}^{\prime \prime }E_{2}^{n}=R^{n}(G\circ F)(A)} and ${\displaystyle {}^{\prime \prime }E_{2}={}^{\prime \prime }E_{\infty }}${\displaystyle {}^{\prime \prime }E_{2}={}^{\prime \prime }E_{\infty }}. On the other hand, by the definition and the lemma,

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

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

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

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

• 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

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

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