Grothendieck spectral sequence on arvau

Wikipedia / 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. Many spectral sequences in algebraic geometry are instances of the Grothendieck spectral sequence, for example the Leray spectral sequence.

For a continuous map there is the (left-exact) direct image. We also have the global section functors

, 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; e.g., a scheme. Then

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

-modules to flasque sheaves,-acyclic. Hence, the hypothesis is satisfied.

We shall use the following lemma:

. We have

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):

be an injective resolution of A, we have:

of the first and the third nonzero terms. By the horseshoe lemma. Hence, we found an injective resolution of the complex:

satisfies the hypothesis of the lemma (cf. the Cartan–Eilenberg resolution.)

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 is G. On the other hand, by the definition and the lemma,