Leray spectral sequence

Question

Let $f : X\to Y$ be a continuous map of topological spaces, $A$ an abelian sheaf on $X$.

We have the Leray spectral sequence

$$E_2^{p,q} := H^p(Y, R^qf_*A)\Rightarrow H^{p+q}(X, A).$$

Could someone please help me get the long exact sequence of the second page right? It is always very unpleasant and literally no reference I have deals with it in a uniform way: everyone has his favorite notation and direction for the differentials and transgression maps.

It should begin with

$$H^0(Y, R^1f_*A)\to H^1(X, A)\to H^1(Y, f_*A)\to ...$$

but what about the higher degree cohomologies and the exact sequence they fit in?

Answer 1

Do you mean the five term exact sequence? That would be the following. $$0 \to H^1(Y,f_*A) \to H^1(X,A) \to H^0(Y,R^1f_*A) \to H^2(Y,f_*A) \to H^2(X,A).$$