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
Many spectral sequences in algebraic geometry are instances of the Grothendieck spectral sequence, for example the Leray spectral sequence.
are two additive and left exact functors between abelian categories such that both
takes injective objects to
there is a spectral sequence: where
denotes the p-th right-derived functor of
' means convergence of spectral sequences.
The exact sequence of low degrees reads If
are topological spaces, let
be the category of sheaves of abelian groups on
there is the (left-exact) direct image functor
We also have the global section functors Then since
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
There is a spectral sequence relating the global Ext and the sheaf Ext: let F, G be sheaves of modules over a ringed space
Then This is an instance of the Grothendieck spectral sequence: indeed, Moreover,
-modules to flasque sheaves,[2] which are
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
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.
, we have: Take injective resolutions
By the horseshoe lemma, their direct sum
Hence, we found an injective resolution of the complex: such that each row
satisfies the hypothesis of the lemma (cf.
gives rise to two spectral sequences, horizontal and vertical, which we are now going to examine.
On the other hand, by the definition and the lemma, Since
(it is a resolution since its cohomology is trivial), Since
have the same limiting term, the proof is complete.
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