Spectral sequence comparison theorem

The inverse of the page map $f_s$ commutes with differentials: multiply $\tilde{d}{f}_s=f_{s}d$ by the componentwise inverses at the source and target to obtain $d{f}_s^{-1}=f_s^{-1}\tilde{d}$. Hence both maps preserve cycle kernels and incoming boundary images. They induce mutually inverse homology quotient maps. The transition identity gives $f_{s+1}={\tilde{\alpha }}_{s}H(f_s){\alpha }_s^{-1}$, an isomorphism. Induction proves $f_r$ is an isomorphism at every bidegree for every $r\ge s$.

Fix $(p,q)$. Choose an integer $r\ge s$ beyond the two stationarity bounds for this position in both sequences. Their specified transitions canonically identify these terms with their limiting terms. By step 1.1 the resulting limiting map $f_{\mathrm{\infty }}:E_{p,q}^{\mathrm{\infty }}\to {\tilde{E}}_{p,q}^{\mathrm{\infty }}$ is an isomorphism. Compatibility of $h_{p+q}$ with the abutment data says that its graded map is this map conjugated by the two specified graded identifications. Thus ${\mathrm{gr}}_p{h}_{p+q}$ is an isomorphism. Only finitely many bounds were compared at each fixed position; there is no uniform-collapse hypothesis.

Fix $n$. Step 2.1 proves that the filtered map $h_n$ induces an isomorphism on every graded piece. Apply [F4] to its finite filtrations in the abelian-category case, or to its exhaustive separated complete module filtrations in the other case. It follows that $h_n$ is invertible with filtered inverse. This uses strong convergence as supplied data, and does not invoke any theorem asserting convergence of an unbounded filtered complex. Zero page terms, a zero target, and a single filtration jump are included in [F4]. No AC is introduced. Without the compatibility in step 2.1, the page map would say nothing about the graded map of the specified $h_n$, so that hypothesis cannot be omitted from this argument.