Grothendieck spectral sequence

In mathematics, in the field of homological algebra, the Grothendieck spectral sequence, introduced in 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.

If F :\mathcal{A}\to\mathcal{B} and G :\mathcal{B}\to\mathcal{C} are two additive and left exact functors between abelian categories such that F takes F-acyclic objects (e.g., injective objects) to G-acyclic objects and if \mathcal{B} has enough injectives, then there is a spectral sequence for each object A of \mathcal{A} that admits an F-acyclic resolution:

E_2^{pq} = ({\rm R}^p G \circ{\rm R}^q F)(A) \Longrightarrow {\rm R}^{p+q} (G\circ F)(A).

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

0 R1G(FA) R1(GF)(A) G(R1F(A)) R2G(FA) R2(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 and
\mathcal{C} = \mathbf{Ab} be the category of abelian groups.

For a continuous map

f : X \to Y

there is the (left-exact) direct image functor

f_* : \mathbf{Ab}(X) \to \mathbf{Ab}(Y).

We also have the global section functors

\Gamma_X : \mathbf{Ab}(X)\to \mathbf{Ab},

and

\Gamma_Y : \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, 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 (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).[1]

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 flaque sheaves,[2] 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 and implies B^{n+1} is injective and the first part of the lemma. Next we look at

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

It splits. Thus,

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

Similarly we have (using the early 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 F-acyclic 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, \bullet}) 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

  1. Godement, Ch. II, Theorem 7.3.3.
  2. Godement, Ch. II, Lemma 7.3.2.

References

This article incorporates material from Grothendieck spectral sequence on PlanetMath, which is licensed under the Creative Commons Attribution/Share-Alike License.

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