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 , from knowledge of the derived functors of and . Many spectral sequences in algebraic geometry are instances of the Grothendieck spectral sequence, for example the Leray spectral sequence.

Statement

edit

If   and   are two additive and left exact functors between abelian categories such that both   and   have enough injectives and   takes injective objects to  -acyclic objects, then for each object   of   there is a spectral sequence:

 

where   denotes the p-th right-derived functor of  , etc., and where the arrow ' ' means convergence of spectral sequences.

Five term exact sequence

edit

The exact sequence of low degrees reads

 

Examples

edit

The Leray spectral sequence

edit

If   and   are topological spaces, let   and   be the category of sheaves of abelian groups on   and  , respectively.

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  .

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 injective 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 1973, Ch. II, Theorem 7.3.3.
  2. ^ Godement 1973, 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. Vol. 38. Cambridge University Press. ISBN 978-0-521-55987-4. MR 1269324. OCLC 36131259.

Computational Examples

edit

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