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. 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. The exact sequence of low degrees reads Leray spectral sequence If and are topological spaces, let and be the on and , respectively. For a continuous map there is the (left-exact) 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 . 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, and . Moreover, sends injective -modules to flasque sheaves, which are -acyclic. Hence, the hypothesis is satisfied. We shall use the following 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.