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

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

Examples

= The Leray spectral sequence =

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

: $E^{p,q}_2 = \operatorname{H}^p(X; \mathcal{E}xt^q_{\mathcal{O}}(F, G)) \Rightarrow \operatorname{Ext}^{p+q}_{\mathcal{O}}(F, G).$ {{harvnb|Godement|1973|loc=Ch. II, Theorem 7.3.3.}}

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

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

Moreover, $\mathcal{H}om_{\mathcal{O}}(F, -)$ sends injective $\mathcal{O}$ -modules to flasque sheaves,{{harvnb|Godement|1973|loc=Ch. II, Lemma 7.3.2.}} which are $\Gamma(X, -)$ -acyclic. Hence, the hypothesis is satisfied.

Derivation

We shall use the following lemma:

{{math_theorem|name=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,

: ${}^{\prime} E^{p, q}_1 = H^q(G(I^{\bullet, p})) = G(H^q(I^{\bullet, p})).$

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),

: ${}^{\prime} E^{p, q}_2 = R^p G(R^qF(A)).$

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

Notes

References

{{Citation | last1=Godement | first1=Roger | author1-link=Roger Godement | title=Topologie algébrique et théorie des faisceaux | publisher=Hermann | location=Paris |mr=0345092 | year=1973}}
{{Weibel IHA}}

= Computational Examples =

Sharpe, Eric (2003). Lectures on D-branes and Sheaves (pages 18–19), {{arxiv|hep-th/0307245}}

{{PlanetMath attribution|id=1095|title=Grothendieck spectral sequence}}

Category:Spectral sequences