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

Statement

If and are two additive and left exact functors between abelian categories such that both \mathcal{A} and \mathcal{B} have enough injectives and F takes injective objects to G-acyclic objects, then for each object A of \mathcal{A} there is a spectral sequence: where {\rm R}^p G denotes the p-th right-derived functor of G, etc., and where the arrow '' means convergence of spectral sequences.

Five term exact sequence

The exact sequence of low degrees reads

Examples

The Leray spectral sequence

If X and Y are topological spaces, let and be the category of sheaves of abelian groups on X and Y, respectively. For a continuous map there is the (left-exact) direct image functor. We also have the global section functors Then since 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: for a sheaf \mathcal{F} of abelian groups on X.

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 ; e.g., a scheme. Then This is an instance of the Grothendieck spectral sequence: indeed, Moreover, sends injective \mathcal{O}-modules to flasque sheaves, which are -acyclic. Hence, the hypothesis is satisfied.

Derivation

We shall use the following lemma: Proof: Let be the kernel and the image of. We have which splits. This implies each B^{n+1} 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. \square We now construct a spectral sequence. Let be an injective resolution of A. Writing \phi^p 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 F(A^p). 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 F(A^p) 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. \square

Computational Examples