Approximate cycle obstruction sequence for a complete filtered complex

Completeness includes injectivity, so ${\bigcap }_m{F}_{-m}{C}_k=0$. Every fixed $F_p{C}_k$ is closed in the residue sense: if $x\in {\bigcap }_t(F_p{C}_k+F_t{C}_k)$, take $t\le p$ to get $x\in F_p{C}_k$. A compatible family of residues defines a unique element of $C_k$ by completeness, and if every sufficiently fine residue is represented in $F_p{C}_k$, its value lies in $F_p{C}_k$ by this same test. Filtration preservation makes application of $d$ compatible with residues.

The transition maps of $A(p,t)$ are inclusions in both coordinates and commute. For fixed $t$ and $p\le t$ one has $A(p,t)=F_p{C}_n$: [F1] gives the forward inclusion in the inverse-image condition, and the reverse is part of the definition. The cofinal $p$ tower therefore has zero limit by separatedness and zero Delta cokernel by [F2] and completeness. The finite-tail identifications of [F3] give $L_{p}A(p,t)=R_{p}A(p,t)=0$ for the full $p$ tower with any fixed upper endpoint.

For the cycle tower at indices $-m$, let $y_m\in Z_{-m}$ be any product tuple. For each $m$ and $l\ge 0$ define a residue modulo $F_{-l}{C}_n$ by the finite sum ${\sum }_{m\le k<l}{y}_k$ when $l>m$, and zero when $l\le m$. The residues are compatible since all newly removed summands lie in the coarser filtration piece. Completeness supplies a unique $x_m\in C_n$. Its residue modulo $F_{-m}$ is zero, so $x_m\in F_{-m}{C}_n$. Applying $d$ to every residue gives zero because each $y_k$ is a cycle; separatedness of $C_{n-1}$ implies $d{x}_m=0$. Thus $x_m\in Z_{-m}$. Comparing the finite sums in every quotient gives $x_m-x_{m+1}=y_m$, since the quotient family separates elements. Delta on this cycle tower is onto, so $R_p{Z}_p=0$, again with arbitrary upper endpoint by [F3].

For fixed $p$, compatibility in the inclusion tower $A(p,t)$ means a single element lies in every $A(p,t)$. By separatedness of $C_{n-1}$ this is precisely $Z_p$. Apply [F4] to the rectangular system restricted to any fixed upper endpoints in $p,t$. Step 2.1 verifies both required vanishings. Thus $L_p{Q}_p=L_p{R}_{t}A(p,t)=0$. Changing endpoints gives the same result by [F3], so this holds for the full minus-infinity tower of the $Q_p$.

Fix $p$ and use a common $t$ endpoint for $A(p-1,t)$ and $A(p,t)$. The kernel of their map to $F_p{C}_n\mathrm{/}{F}_{p-1}{C}_n$ is exactly $A(p-1,t)$; hence $0\to A(p-1,t)\to A(p,t)\to S(p,t)\to 0$ is termwise exact. The maps on $S$ are inclusions of nested submodules in the fixed graded module. Its limit is their intersection: compatibility means every coordinate is the same element. Applying [F3] and step 3.1 identifies the first three limit terms as $Z_{p-1},Z_p,S(p,\mathrm{\infty })$ and gives the asserted sequence, with the $Q$ map induced by inclusion.

A filtration-preserving chain map sends each $A(p,t)$, $Z_p$ and $S(p,t)$ to its counterpart. It commutes with their inclusions and quotient maps and therefore with the six-term sequence by [F3]. The double-Delta comparison is natural by [F4]; the residue construction is compatible because a continuous filtered map sends the uniquely determined residues to their images. This proves the stated naturality.

Steps 3.1, 4.1 and 2.2 establish the sequence and both vanishings. Zero chain groups give zero towers and zero sequences. A finite lower filtration bound makes all sufficiently small $A(p,t)$ in the $p$ direction zero, consistently with the argument. Repeated pieces and zero differentials are allowed: for $d=0$, $A(p,t)=Z_p=F_p{C}_n$ is constant in $t$, so $Q_p=0$ and the exact sequence reduces to the graded quotient sequence. No sum over an unbounded set of nonvanishing residues was taken: each residue in step 2.2 is a finite sum, including the empty sum at $l\le m$. AC is confined to the cited tower lemmas as declared in [F5].