Grothendieck 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 ${\displaystyle G\circ F}$, from knowledge of the derived functors of F and G.

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

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

where ${\displaystyle {\rm {R}}^{p}G}$ denotes the p-th right-derived functor of ${\displaystyle G}$, etc.

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

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

Examples

The Leray spectral sequence

If ${\displaystyle X}$ and ${\displaystyle Y}$ are topological spaces, let

${\displaystyle {\mathcal {A}}=\mathbf {Ab} (X)}$ and ${\displaystyle {\mathcal {B}}=\mathbf {Ab} (Y)}$ be the category of sheaves of abelian groups on X and Y, respectively and

${\displaystyle {\mathcal {C}}=\mathbf {Ab} }$ be the category of abelian groups.

For a continuous map

${\displaystyle f\colon X\to Y}$

there is the (left-exact) direct image functor

${\displaystyle f_{*}\colon \mathbf {Ab} (X)\to \mathbf {Ab} (Y)}$.

We also have the global section functors

${\displaystyle \Gamma _{X}\colon \mathbf {Ab} (X)\to \mathbf {Ab} }$,

and

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

Then since

${\displaystyle \Gamma _{Y}\circ f_{*}=\Gamma _{X}}$

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

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

for a sheaf ${\displaystyle {\mathcal {F}}}$ of abelian groups on ${\displaystyle X}$, and this is exactly the Leray spectral sequence.

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 ${\displaystyle (X,{\mathcal {O}})}$; e.g., a scheme. Then

${\displaystyle E_{2}^{p,q}=\operatorname {H} ^{p}(X;{\mathcal {E}}xt_{\mathcal {O}}^{q}(F,G))\Rightarrow \operatorname {Ext} _{\mathcal {O}}^{p+q}(F,G).}$[1]

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

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

Moreover, ${\displaystyle {\mathcal {H}}om_{\mathcal {O}}(F,-)}$ sends injective ${\displaystyle {\mathcal {O}}}$-modules to flasque sheaves,[2] which are ${\displaystyle \Gamma (X,-)}$-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,

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

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

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

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

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

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

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

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

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

Similarly we have (using the earlier splitting):

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

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

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

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

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

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

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

Now, the double complex ${\displaystyle E_{0}^{p,q}=G(I^{p,q})}$ gives rise to two spectral sequences, horizontal and vertical, which we are now going to examine. On the one hand, by definition,

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

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

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

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

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

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

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, MR 0345092
• Weibel, Charles A. (1994). An introduction to homological algebra. Cambridge Studies in Advanced Mathematics. 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

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