Grothendieck spectral sequence

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

$E_2^{pq}=({\mathrm{R}}^{p}G\circ {\mathrm{R}}^{q}F)(A)⟹{\mathrm{R}}^{p+q}(G\circ F)(A),$

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

Five term exact sequence

The exact sequence of low degrees reads

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

Examples

The Leray spectral sequence

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

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

${\mathrm{\Gamma }}_X:\mathbf{𝐀}\mathbf{𝐛}(X)\to \mathbf{𝐀}\mathbf{𝐛}$ and ${\mathrm{\Gamma }}_Y:\mathbf{𝐀}\mathbf{𝐛}(Y)\to \mathbf{𝐀}\mathbf{𝐛}.$

Then since ${\mathrm{\Gamma }}_Y\circ f_{*}={\mathrm{\Gamma }}_X$ and the functors $f_{*}$ and ${\mathrm{\Gamma }}_Y$ 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:

$H^p(Y,{\mathrm{R}}^q{f}_{*}{ℱ})⟹H^{p+q}(X,{ℱ})$

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

$E_2^{p,q}={\mathrm{H}}^p(X;{ℰ}x{t}_{{𝒪}}^q(F,G))\Rightarrow {\mathrm{Ext}}_{{𝒪}}^{p+q}(F,G).$

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

$R^p\mathrm{\Gamma }(X,-)={\mathrm{H}}^p(X,-)$, $R^q{\mathscr{H}}o{m}_{{𝒪}}(F,-)={ℰ}x{t}_{{𝒪}}^q(F,-)$ and $R^n\mathrm{\Gamma }(X,{\mathscr{H}}o{m}_{{𝒪}}(F,-))={\mathrm{Ext}}_{{𝒪}}^n(F,-)$.

Moreover, ${\mathscr{H}}o{m}_{{𝒪}}(F,-)$ sends injective ${𝒪}$-modules to flasque sheaves, which are $\mathrm{\Gamma }(X,-)$-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

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

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):

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

The second part now follows. $◻$

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

$0\to \ker {\varphi }^p\to F(A^p)\stackrel{{\varphi }^p}{\to }\mathrm{im}{\varphi }^p\to 0.$

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 ${}^{\prime \prime }{E}_2={}^{\prime \prime }{E}_{\mathrm{\infty }}$. 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. $◻$

References

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

Computational Examples

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