I have a hopefully simple question about the Hochschild-Serre spectral sequence (which may just be a simple question about general spectral sequences). Whenever I see the sequence written down, it has always been in the following form:
Let $G$ be a group with normal subgroup $N$ and $V$ a $G$-module. Then we have the convergent cohomological spectral sequence
$$E_2^{pq} = H^p(G/N, H^q(N,V)) \Rightarrow H^{p+q}(G,V).$$
Does this then mean we have a filtration of $H^n(G,V)$ by those $E_2^{pq}$ such that $p+q = n$? For example, if we apply this for $n=2$ do we then have that $H^2(G,V)$ has composition factors (comprising the composition factors of) $H^2(G/N, V^N)$, $H^1(G/N, H^1(N,V))$ and $H^2(N,V)^{G/N}$?
I only ask since when looking at this spectral sequence one also often finds the 5-term exact sequence of low-degree terms (also the inflation-restriction exact sequence)
$$0 \to H^1(G/N, V^N) \to H^1(G,V) \to H^1(N,V)^{G/N} \xrightarrow{d} H^2(G/N, V^n) \to H^2(G,V)$$
and if the above claim is true then is $d$ not always forced to be $0$ in this exact sequence (since then $H^1(G,V)$ would be an extension of $H^1(N,V)^{G/N}$ by $H^1(G/N,V^n)$)?
When you have a convergent spectral sequence $$ E_2^{pq}\Rightarrow H^{p+q} $$ you indeed have a filtration on $H^{pq}$, but its graded part are $E_{\infty}^{pq}$ which are subquotients of $E_2^{pq}$. At each page of the spectral sequence, some elements might not survive to the next page. The $E_\infty^{pq}$ are those who survive the whole sequence.
As a matter of fact, there are spectral sequences with a non zero $E_2$ page but whose abutment is $H^n=0$.
Let us see where the 5-terms low degree exact sequence comes from. This is essentially the study of the first non trivial differential at $E_2$, namely : $E_2^{10}\rightarrow E_2 ^{20}$.
When you have a first quadrant spectral sequence, elements of $E_2^{10}$ cannot be killed anymore (look at the differentials from and to $E_2^{10}$) so they survive the sequence and inject into $H^1$. This is the first step in the filtration and we want to understand the quotient.
You have a map $H^1\rightarrow E_2^{01}$ whose kernel is $E_2^{10}$, but this time, $E_2^{01}$ is too big : there is a differential $E_2^{01}\rightarrow E_2^{20}$ which might not be zero. So every element which is not a cocyle does not survive the spectral sequence. The subset of cocycle is by definition $E_3^{01}=E_{\infty}^{01}$ (because there won't be any differential anymore). So the graduation of $H^1$ is $$0\rightarrow E^{10}_2\rightarrow H^1\rightarrow E_3^{01}\rightarrow 0$$ with $E_3^{01}=\ker(E_2^{10}\rightarrow E_2^{20})$.
Similarily, elements of $E_2^{20}$ does not necessarily survive through the spectral sequence because some of them are boundary of $E_2^{10}\rightarrow E_2^{20}$. The quotient is $E_3^{20}=E_\infty^{20}$ (since there won't be any differential anymore) which inject into $H^2$. So you have a sequence $$0\rightarrow E_3^{01}\rightarrow E_2^{10} \rightarrow E_2^{20}\rightarrow E_3^{20}\rightarrow 0$$ Putting the above two exact sequences and $0\rightarrow E_3^{20}\rightarrow H^2$ together, you got the five terms sequence of low degree.