typically expresses something vague like “the spectral sequence computes \(A_*\)”. But it is not always clear that it does! In practice, solving the convergence problem amounts to relating \(A_*\) (the abutment) to the limit page \(E^\infty_{*,*}\text{.}\) The strongest form of this relationship is strong convergence.
If the filtration is finite for each \(n\) (i.e. \(F_s A_n = 0\) for \(s\) small enough and \(F_s A_n = A_n\) for \(s\) large enough), then exhaustiveness, Hausdorffness, and completeness are automatic.
The isomorphisms \(E^\infty_{p,q} \cong F_p A_{p+q} / F_{p-1} A_{p+q}\) are the best we can do to recover \(A_{p+q}\) from the spectral sequence: one obtains \(A_{p+q}\) only up to the extension problem of reassembling its filtration quotients. The cohomological grading uses the dual convention of a decreasing filtration \(\dots \supseteq F^{s+1} A^n \supseteq F^s A^n \supseteq \dots\) with \(E_\infty^{p,q} \cong F^p A^{p+q} / F^{p+1} A^{p+q}\text{.}\)
