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 , from knowledge of the derived functors of F and G.

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

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 and are topological spaces, let

and be the category of sheaves of abelian groups on X and Y, respectively and
be the category of abelian groups.

For a continuous map

there is the (left-exact) direct image functor

.

We also have the global section functors

,

and

Then since

and the functors and satisfy the hypotheses (since the direct image functor has an exact left adjoint , pushforwards of injectives are injective and in particular acyclic for the global section functor), the sequence in this case becomes:

for a sheaf of abelian groups on , 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 ; e.g., a scheme. Then

[1]

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

, and .

Moreover, sends injective -modules to flasque sheaves,[2] which are -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,

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

Proof: Let be the kernel and the image of . We have

which splits. This implies each is injective. Next we look at

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

Similarly we have (using the earlier splitting):

The second part now follows.

We now construct a spectral sequence. Let be an F-acyclic resolution of A. Writing for , we have:

Take injective resolutions and of the first and the third nonzero terms. By the horseshoe lemma, their direct sum is an injective resolution of . Hence, we found an injective resolution of the complex:

such that each row satisfies the hypothesis of the lemma (cf. the Cartan–Eilenberg resolution.)

Now, the double complex gives rise to two spectral sequences, horizontal and vertical, which we are now going to examine. On the one hand, by definition,

,

which is always zero unless q = 0 since is G-acyclic by hypothesis. Hence, and . On the other hand, by the definition and the lemma,

Since is an injective resolution of (it is a resolution since its cohomology is trivial),

Since and have the same limiting term, the proof is complete.

Notes[edit]

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

References[edit]

  • Godement, Roger (1973), Topologie algébrique et théorie des faisceaux, Paris: Hermann, MR 0345092
  • Weibel, Charles A. (1994). An introduction to homological algebra. Cambridge Studies in Advanced Mathematics. 38. Cambridge University Press. ISBN 978-0-521-55987-4. MR 1269324. OCLC 36131259.
Computational Examples[edit]
  • Shape, Eric (2003). Lectures on D-branes and Sheaves (pages 18-19), Arxiv

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