let me say first that I don't know homological algebra very well, so I apologize in advance if my question is stupid..
It regards the Serre spectral sequence associated to a fibration $0\rightarrow F \rightarrow_i E \rightarrow_p B\rightarrow 0$.
I know it is such that $E_2^{p,q}=H^p(B,H^q(F,\mathbb{C}))$ converges to $E_{\infty}^{p,q}=H^{p+q}(E,\mathbb{C})$.
I've read that, given $H^1(F,\mathbb{C})=0$, from this sequence i can derive an exact sequence
$0\rightarrow H^2(B,\mathbb{C})=E_2^{2,0}\rightarrow_{e_B} H^2(E,\mathbb{C})=E_{\infty}^{2,0}\rightarrow_{e_F} H^2(F,\mathbb{C})=E_2^{0,2}$
where $e_B$ is the morphism induced by the inclusion $i:F\rightarrow E$ and which is realized at the level of spectral sequences by the sequence $E_2^{2,0}\rightarrow E_3^{2,0} \rightarrow \ldots\rightarrow E_{\infty}^{2,0}$ while $e_F$ is the morphism induced by the projection $p:E\rightarrow B$ and which is realized at the level of spectral sequences by the sequence $E_2^{2,0}=E_2^{0,2}\rightarrow E_3^{0,2} \rightarrow \ldots\rightarrow E_{\infty}^{0,2}$.
Now my problem is of course that i can not see how this is an exact sequence.. because $E_3^{2,0}=Ker (E_2^{2,0}\rightarrow E_2^{4,-1})/Im(E_2^{0,1}\rightarrow E_2^{2,0})=E_2^{2,0}$ and so on $E_r^{2,0}=E_2^{2,0}$.
Also, i think this is a general theorem on exact sequences such that $E_2^{p,q}\Rightarrow H^{p+q}(Tot(M))$: if $E_2^{p,q}=0$ for every $q<n$ then there is
$0\rightarrow E_2^{p,0}\rightarrow H^{p+q}(Tot(M))\rightarrow E_2^{0,p}\rightarrow\ldots$
So would you point me out the proof or a reference where i can find it?
Thank you.
It is a long exact sequence because it can be built up from kernel cokerel sequences which are obviously exact. see mosher and tangora's chapter on fiber spaces. Alternatively see section 4 after proposition 2, in serre's singular homology of fiber spaces.