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 $G\circ F$, 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.

Statement

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

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

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

Five term exact sequence

The exact sequence of low degrees reads

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

Examples

The Leray spectral sequence

Main article: Leray spectral sequence

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

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

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

Then since $\Gamma _{Y}\circ f_{*}=\Gamma _{X}$ and the functors $f_{*}$ and $\Gamma _{Y}$ 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:

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

Similarly we have (using the earlier splitting):

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

The second part now follows. $\square$

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

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

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

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

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

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

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

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

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

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

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

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

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. 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.