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

Statement

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

[math]\displaystyle{ E_2^{pq} = ({\rm R}^p G \circ{\rm R}^q F)(A) \Longrightarrow {\rm R}^{p+q} (G\circ F)(A), }[/math]

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

Five term exact sequence

The exact sequence of low degrees reads

[math]\displaystyle{ 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). }[/math]

Examples

The Leray spectral sequence

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

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

[math]\displaystyle{ \Gamma_X \colon \mathbf{Ab}(X)\to \mathbf{Ab} }[/math] and [math]\displaystyle{ \Gamma_Y \colon \mathbf{Ab}(Y) \to \mathbf {Ab}. }[/math]

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

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

for a sheaf [math]\displaystyle{ \mathcal{F} }[/math] of abelian groups on [math]\displaystyle{ X }[/math].

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 [math]\displaystyle{ (X, \mathcal{O}) }[/math]; e.g., a scheme. Then

[math]\displaystyle{ 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). }[/math]

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

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

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

[math]\displaystyle{ H^n(K^{\bullet}) }[/math]

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

[math]\displaystyle{ H^n(G(K^{\bullet})) = G(H^n(K^{\bullet})). }[/math]

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

[math]\displaystyle{ 0 \to Z^n \to K^n \overset{d}\to B^{n+1} \to 0, }[/math]

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

[math]\displaystyle{ 0 \to B^n \to Z^n \to H^n(K^{\bullet}) \to 0. }[/math]

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

[math]\displaystyle{ 0 \to G(B^n) \to G(Z^n) \to G(H^n(K^{\bullet})) \to 0. }[/math]

Similarly we have (using the earlier splitting):

[math]\displaystyle{ 0 \to G(Z^n) \to G(K^n) \overset{G(d)} \to G(B^{n+1}) \to 0. }[/math]

The second part now follows. [math]\displaystyle{ \square }[/math]

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

[math]\displaystyle{ 0 \to \operatorname{ker} \phi^p \to F(A^p) \overset{\phi^p}\to \operatorname{im} \phi^p \to 0. }[/math]

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

[math]\displaystyle{ 0 \to F(A^{\bullet}) \to I^{\bullet, 0} \to I^{\bullet, 1} \to \cdots. }[/math]

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

Now, the double complex [math]\displaystyle{ E_0^{p, q} = G(I^{p, q}) }[/math] gives rise to two spectral sequences, horizontal and vertical, which we are now going to examine. On the one hand, by definition,

[math]\displaystyle{ {}^{\prime \prime} E_1^{p, q} = H^q(G(I^{p, \bullet})) = R^q G(F(A^p)) }[/math],

which is always zero unless q = 0 since [math]\displaystyle{ F(A^p) }[/math] is G-acyclic by hypothesis. Hence, [math]\displaystyle{ {}^{\prime \prime} E_{2}^n = R^n (G \circ F) (A) }[/math] and [math]\displaystyle{ {}^{\prime \prime} E_2 = {}^{\prime \prime} E_{\infty} }[/math]. On the other hand, by the definition and the lemma,

[math]\displaystyle{ {}^{\prime} E^{p, q}_1 = H^q(G(I^{\bullet, p})) = G(H^q(I^{\bullet, p})). }[/math]

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

[math]\displaystyle{ {}^{\prime} E^{p, q}_2 = R^p G(R^qF(A)). }[/math]

Since [math]\displaystyle{ {}^{\prime} E_r }[/math] and [math]\displaystyle{ {}^{\prime \prime} E_r }[/math] have the same limiting term, the proof is complete. [math]\displaystyle{ \square }[/math]

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