In the above situation, assume that for every injective object the object is a -acyclic object.
Then for every object there is a spectral sequence called the Grothendieck spectral sequence whose -page is the composite
of the right derived functors of and in degrees and , respectively and which is converging to to the derived functors of the composite of and :
Moreover, this is natural in .
By assumption of enough injectives, we may find an injective resolution
of . Next, by the discussion at injective resolution – Existence and construction we may find a fully injective resolution of the chain complex :
where hence is a double complex of injective objects such that for each the component is an ordinary injective resolution of .
Thus we have the corresponding double complex in . The claim is that the Grothendieck spectral sequence is the spectral sequence of a double complex for equipped with the vertical-degree filtration :
To see this, notice that by the assumption that is a fully injective projective resolution, the short exact sequences
are split (by the discussion there) and hence so is their image under any functor and hence in particular under . Accordingly we have
(the first two equivalences by general properties of the filtration spectral sequence, the last by the above splitness). Hence it follows that
where in the last step we used that is be construction an injective resolution of (using the -acyclicity of ).
This establishes the spectral sequence and its second page as claimed. It remains to determine its convergence.
To that end, conside dually, the spectral sequence coming from the horizontal filtration on the double complex . By the general properties of spectral sequence of a double complex this converges to the same value as the previous one. But for this latter spectral sequence we find
the first equivalence by the general properties of filtration spectral sequences, the second then by the definition of right derived functors. But by assumption is -acyclic and hence all these derived functors vanish in positive degree, so that
Next, the -page then contains just horizontal homology of and this is by definition now the derived functor of the composite of with :
Since this is concentrated in the -row the spectral sequence of the horizontal filtration collapses here and hence
So in conclusion we have
Many other classes of spectral sequences are special cases of the Grothendieck spectral sequence, for instance the