I am looking for a reference for what is stated in Srinivasan's book "Representations of Finite Chevalley Groups", which is apparently a special case of Leray spectral sequence. I'll quote the interesting part : 



> 
> Let $\phi$ be a torsion sheaf on $X$,
>  $f : X \to Y$ a morphism of schemes.
>  Then we have a Leray spectral sequence
>  $ E\_2^{pq} = H^p\_c(Y, R^q f\_{!} \phi )
> > \Rightarrow H^{p+q}\_c(X, \phi)$. The
>  way we use this is as follows.
> 
> 
> Suppose that all the fibres of $f$ are
>  isomorphic to a fixed scheme $Z$ such
>  that $H^q\_c(Z, \phi) = 0$ except for
>  $q= q\_0$. Then $H^p\_c(Y, R^{q\_0} f\_{!}
> > \phi ) \simeq H^{p+q}\_c(X, \phi)$.
> 
> 
> 


The reference I'm looking for is for the second part of what I quoted. 


Since I just happened to work with spectral sequences, maybe this statement is obviously equivalent to the definition of convergence of spectral sequences, in which case I'm sorry for asking.



Special case of Leray spectral sequence
The assumption on the fibers is equivalent to that $R^qf\_! \phi$ vanishes unless $q = q\_0$. Hence there is only one nonzero row in the $E\_2$ page of the spectral sequence, which happens so often that there is a special name for this situation: the spectral sequence *collapses* at $E\_2$. Collapsing is stronger than both degenerating and converging. In particular, if you know that a spectral sequence collapses, then you can read off the actual cohomology groups that the sequence converges to and not just their respective associated gradeds.


All this follows because no maps after the $E\_1$ page send the nonzero row to itself, so there can be no nonzero differentials; moreover, the filtration on each $H^i$ of the $H^\ast$ that the sequence converges to has at most one step (corresponding to the nonzero row), so each $H^i$ is simply given by the corresponding entry in the nonzero row. This should be explained in any reference on spectral sequence, e.g. Weibel's "An introduction to homological algebra".


