Grothendieck spectral sequence

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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[edit]

The Leray spectral sequence[edit]

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[edit]

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^{p,q}_2 = \operatorname{H}^p(X; \mathcal{Ext}^q_O(F, G)) \Rightarrow \operatorname{Ext}^{p+q}_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{Hom}_O(F, -) = \mathcal{Ext}^q_O(F, -) and R^n \Gamma(X, \mathcal{Hom}_O(F, -)) = \operatorname{Ext}^n_O(F, -).

Moreover, \mathcal{Hom}_O(F, -) sends injective O-modules to flaque sheaves,[2] which are \Gamma(X, -)-acyclic. Hence, the hypothesis is satisfied.

Derivation[edit]

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[edit]

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

References[edit]

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