I am reading the Rotman's Introduction to homological algebra, p.641, Proposition 10.28 and stuck at understanding some statement :
I am trying to understand the underlined statements. For example,
Q. 1) Why $E^{\infty}_{1, n-1} = Z^{\infty}_{1,n-1}/B^{\infty}_{1,n-1} =0$ (c.f. his book p.624 definition) is true?
Q. 2) Why $\Phi^{t+1}H_n / \Phi^{t}H_n \cong E^{\infty}_{t+1,n-t-1}=0$ (If so, then $\Phi^{t}H_n = \Phi^{t+1}H_n $) is true?
What property of $E^{\infty}$ can we use? Using the (ii) in the proposition 10.28? I think that I am unfamiler to deep(?) structure of the limit term $E^{\infty}$ of spectral sequence. (If we understand this issue, then it seems that it will be great help for increasing undersanding theory of spectral sequence.) Can anyone helps?
Observe that $E^\infty_{p,q}$ is a subquotient of $E^2_{p,q}$. By hypothesis, $E^2_{p,q} = 0$ if $p \notin \{0, t\}$. So $E^\infty_{p,q} = 0$ for $p \notin \{0, t\}$.