I have a first-quadrant spectral sequence $E^r_{p, q}$ of abelian groups of finite rank converging to $E^{\infty}_{p, q}$. We have $E^{\infty}_{p, q}=E^{\infty}_{r, s}$ if $p+q=r+s$. We also have $E^2_{p, 0}$ abstractly isomorphic to $E^{\infty}_{p, 0}$ for all $p$. Does it mean that the spectral sequence degenerates at the second sheet?
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2$\begingroup$ As written, the answer is no, as you could have $E^\infty_{p,q},E^2_{p,0}\cong 0$ for all $p,q\geq 0$, but still have a nonzero differential (perhaps an isomorphism) between groups not on the bottom row. But judging by your title, maybe you wanted to ask something slightly different? $\endgroup$– Mark GrantCommented Dec 3, 2018 at 10:23
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