Cohomological Serre spectral sequence

Apply [F1] to $K^{\ast }=C^{\ast }(E;R)$ with the decreasing filtration [F3]. It gives $E_1^{a,b}=H^{a+b}(E_a,E_{a-1};R)$ and differentials of bidegree $(r,1-r)$. No convergence clause of [F1] is invoked yet.

We first prove the relative vanishing needed for convergence. Let $m\ge 0$, $0\le k\le m$, and let $c$ be a finite relative integral $k$-cycle for $(E,E_m)$. Realize its finitely many simplices and faces by [F4] on a finite CW complex $K$ of dimension at most $k$; the support of $\mathrm{\partial }c$ generates a subcomplex $L\subseteq K$ mapping into $E_m$. Apply finite cellular approximation to $pv{\mathrm{\mid }}_L:L\to B^m$. Extend that base homotopy over $K$ by the cofibration clause in [F4] and lift the extension starting at $v:K\to E$. The endpoint is cellular on $L$. Now apply relative cellular approximation to $(K,L)\to (B,B^m)$ rel $L$ and lift it rel $L$. The final projection is cellular, so its image lies in $B^k\subseteq B^m$, while the combined homotopy of $L$ stays in $B^m$. The prism identity makes $c$ equal, modulo a boundary and a chain in $E_m$, to a chain entirely in $E_m$. Thus $H_k(E,E_m;\mathbb{Z})=0(0\le k\le m)\mathrm{.}$ Every construction is finite and this step itself uses no choice.

Fix an oriented $a$-cell. Use the same finite lifted disk and nested hemispheres as [F2], but apply cohomology contravariantly. Excision and the pair sequences in [F2] give, by $a$ successive positive connecting maps, $H^b(F_{x_e};R)\stackrel{\sim }{\to }{H}^{a+b}(p^{-1}(\bar{e}),p^{-1}(\mathrm{\partial }e);R)\mathrm{.}$ For $a=0$ this is the identity. At each suspension step the two endpoint restrictions have diagonal image; its cokernel identifies the two endpoint coordinate maps with opposite signs, fixing the orientation sign. To separate all cells, use [F2]'s one uniform radial collar, enlarge $E_{a-1}$ to the inverse image of the outer collar, and excise a smaller closed collar. The remaining pair is the disjoint union of the pulled-back concentric cell pairs. Every singular simplex in this union lies in one component, so its relative integer chain complex is the direct sum of the component complexes; [F3] therefore identifies its relative cochain complex with their product. Kernels are coordinatewise, and [A1] lets one choose a primitive in every nonempty coordinate primitive set, so the image of the product coboundary is the product of its images. Cohomology consequently splits as the product, giving $E_1^{a,b}\cong C_{\mathrm{c}\mathrm{e}\mathrm{l}\mathrm{l}}^a(B;{\mathcal{H}}^b(p;R))\mathrm{.}$ The reversed base path in the coefficient system is exactly the contravariant fiber map used by these connectors.

By [F3], $F^{m+1}{K}^{\ast }=C^{\ast }(E,E_m;R)$. Its integral relative chain groups are free on the singular simplices not lying in $E_m$. Under [A1], the UCT in [F5] has outer terms ${\mathrm{Ext}}_{\mathbb{Z}}^1(H_{k-1}(E,E_m;\mathbb{Z}),R),{\mathrm{Hom}}_{\mathbb{Z}}(H_k(E,E_m;\mathbb{Z}),R)\mathrm{.}$ Both vanish for $k\le m$ by Step 1.2, so $H^k(F^{m+1}{K}^{\ast })=H^k(E,E_m;R)=0(k\le m)\mathrm{.}$ This is the only convergence step that uses UCT and it explicitly carries [A1].

Naturality of the cohomological pair connectors and excision maps reduces $d_1$ to the attaching incidences. The reflected interval calculation in Step 2.1 gives the negative incidence sign, and contravariance reverses the fiber path, so the resulting matrix is precisely the cellular local-coefficient coboundary. The cellular comparison in [F2] therefore gives $E_2^{a,b}\cong H^a(B;{\mathcal{H}}^b(p;R))\mathrm{.}$ Since $E_1$ is first quadrant, every later page is first quadrant.

Fix total degree $n\ge 0$ and set $N=2n+3$. Step 2.2 gives $H^n(F^{N}K)=H^{n+1}(F^{N}K)=0$. Apply the long exact sequence in [F5] to $0⟶F^{N}K⟶K⟶K\mathrm{/}{F}^{N}K⟶0$ and also to $0\to F^{N}K\to F^{a}K\to F^{a}K\mathrm{/}{F}^{N}K\to 0$ for $0\le a\le n+1$. It follows that $H^n(K)\to H^n(K\mathrm{/}{F}^{N}K)$ is an isomorphism identifying every image-filtration term $F^a{H}^n$. The quotient filtration has finite endpoints $F^0=K\mathrm{/}{F}^{N}K$ and $F^N=0$, so [F1] gives its natural finite abutment.

For a square over a cellular $f$, one has $u(E_a)\subseteq E_a^{\mathrm{\prime }}$. Precomposition therefore sends a cochain vanishing on $E_{a-1}^{\mathrm{\prime }}$ to one vanishing on $E_{a-1}$, so $u^{\ast }$ is a filtered cochain map. Functoriality in [F1] gives the contravariant maps on all pages. The cellwise constructions in Steps 2.1–3.1 are natural, and [F6] identifies the $E_2$ map with $f^{\ast }$ for the coefficient morphism $f^{\ast }{\mathcal{H}}^b(p^{\mathrm{\prime }};R)\to {\mathcal{H}}^b(p;R)$.

At a position $(a,b)$ with $a+b=n$, both incident cohomological differentials vanish once $r>\max (a,b+1)$: the incoming source then has negative first coordinate, and the outgoing target has negative second coordinate. Put $s=\max (a,b+1)+1\le n+2$. Under the reindexing in [F1], quotienting by $F^{N}K$ removes a chain-filtration piece below the finite window used through page $s$, since $N=2n+3\ge a+s+1$. The finite-window clause of [F5] therefore identifies the original and quotient pages through their stationary page at $(a,b)$. The quotient's finite abutment from Step 3.2 consequently gives the specified natural identification $E_{\mathrm{\infty }}^{a,b}\cong F^a{H}^n(E;R)\mathrm{/}{F}^{a+1}{H}^n(E;R)\mathrm{.}$

The definition gives $F^{0}K=K$, hence $F^0{H}^n=H^n$. Taking $m=n$ and $k=n$ in Step 2.2 gives $H^n(F^{n+1}K)=0$, hence $F^{n+1}{H}^n=0$. Thus the target filtration has finite endpoints and is exhaustive, separated, and complete by [F5]. Step 4.2 proves two-sided regularity and the actual associated-graded identifications, so all strong-convergence conditions hold. The quotient comparisons and actual cocycle maps are functorial, so these identifications are compatible with Step 4.1.

If $B=\mathrm{\varnothing }$, then $E=\mathrm{\varnothing }$ and all groups vanish. Empty fibers give zero stalks, and the zero ring gives zero cochains. The cases $a=0$, $b=0$, $n=0$, a single cell, constant or degenerate simplices, repeated filtration terms, and the first/last target pieces occur in Steps 1.1–5.1. Step 4.2 checks both incident differential bounds, and Step 5.1 checks both finite endpoints and both associated-graded directions. All product-of-images and UCT uses cite [A1]; finite cellular deformations do not. The theorem has no iff assertion.