Spectral sequence of a tower

Set $D_{p,q}^1={\pi }_{p+q}{F}^{-p}E$ and $E_{p,q}^1={\pi }_{p+q}{\mathrm{gr}}^{-p}E$. Let $i$ be induced by $F^{-p}E\to F^{-p-1}E$, let $j$ be the stage projection, and let $k$ be the connecting homomorphism. Their respective degrees are $(1,-1)$, $(0,0)$ and $(-1,0)$. Exactness at all three vertices follows from the long exact sequences in [F1], so these maps form an initial exact couple.

Derive the couple repeatedly. At homological page $r\ge 1$, its differential $d_r=jk$ has degree $(-r,r-1)$, squares to zero, and its homology is the next page. Define $E_s^{a,b}=E_{b,-a-2b}^{s-1}$ for $s\ge 2$. A homological differential with $r=s-1$ changes $(b,-a-2b)$ to $(b-r,-a-2b+r-1)$, corresponding to $(a+s,b-s+1)$. Hence these reindexed pages satisfy the cohomological spectral-sequence convention. At $s=2$, their total homotopy degree is $b+(-a-2b)=-a-b$ and their filtration index is $-b$, proving the first displayed formula.

With $j=-b$ and $n=-a-b$, [F3] gives ${\pi }_n{\mathrm{gr}}^{j}E\cong H^{2j-n}(C_j)=H^{a-b}(C_{-b})$. This proves the second displayed formula and its naturality for maps preserving the given graded equivalences.

Fix $j,n$. Iterating the derived-couple formulas shows the following two elementary lifting facts. A class on the initial graded term that survives through arbitrarily many differentials has connecting image in ${\pi }_{n-1}{F}^{j+1}E$ admitting lifts from arbitrarily high stages; the indeterminacies in choosing lifts are precisely the boundaries already factored out. Conversely, if its connecting image is zero, exactness writes it as the stage projection of an element of ${\pi }_n{F}^{j}E$, and that projected class survives every page. The first assertion follows inductively because passing to the next derived couple replaces the $D$-term by the image of the next transition map; the converse follows because a stage projection has zero connecting image and the same is true after each derivation.

The high-stage bound in degree $n-1$ makes the liftable subgroup in step 4.1 zero after finitely many stages. Thus every eventual class is represented by the stage projection of some $x\in {\pi }_n{F}^{j}E$. The derived-couple formulas also give the boundary test: such a projection is eventually zero exactly when the image of $x$ in a sufficiently low stage lies in the image there of ${\pi }_n{F}^{j+1}E$. Indeed, a boundary coming from a lower graded stage has connecting image in ${\pi }_n{F}^{j}E$ whose image in that lower stage vanishes; conversely, if $x$ differs by an element from $F^{j+1}$ by an element killed at a lower stage, repeated exactness lifts that difference through the intervening connecting maps and expresses its projection as an iterated boundary. These implications use only finitely many stages.

Choose the low stage at or below $L_n$. Its map to ${\pi }_{n}E$ is an isomorphism, so the boundary test says precisely that the image of $x$ in ${\pi }_{n}E$ belongs to $F^{j+1}{\pi }_{n}E$. Consequently the projection induces an isomorphism from $F^j{\pi }_{n}E\mathrm{/}{F}^{j+1}{\pi }_{n}E$ to the eventual graded term. The high- and low-stage bounds also bound the possible sources and targets of differentials at the fixed bidegree, so this eventual term is reached after finitely many pages. Finally $F^j{\pi }_{n}E={\pi }_{n}E$ for $j\le L_n$ and $F^j{\pi }_{n}E=0$ for $j\ge U_n$; hence the image filtration is finite and the stated convergence follows. Substitution of $j=-b$ and $n=-a-b$ gives the displayed indexing.