Serre filtration over the base skeleta

Let $p:E\to B$ be a continuous map to a CW complex with skeleta $B^a$. Put $B^a=\mathrm{\varnothing }$ for $a<0$ and $E_a=p^{-1}(B^a)\mathrm{.}$ For a commutative ring $R$, the homological Serre filtration on singular chains is the increasing filtration $F_a{C}_n(E;R)=\mathrm{im}(C_n(E_a;R)\to C_n(E;R))=C_n(E_a;R),F_{-1}{C}_n=0.$ The last equality identifies a singular simplex in the subspace with the same simplex in $E$. Since faces remain in the same preimage, $\mathrm{\partial }{F}_a{C}_n\subseteq F_a{C}_{n-1}$.

This filtration is exhaustive one chain at a time. Indeed, the projection to $B$ of each singular simplex has compact domain, so Compact CW images have finite cell support without choice places its image in a finite subcomplex and hence in some skeleton. A finite chain has one maximum of its finitely many resulting dimensions. There is generally no uniform bound depending only on $n$: an $n$-simplex can map into cells of arbitrarily high dimension. Thus this definition does not assert that $F_n{C}_n=C_n$ or that the chain filtration is degreewise finite.

The induced increasing filtration on homology is $F_a{H}_n(E;R)=\mathrm{im}(H_n(E_a;R)\to H_n(E;R))\mathrm{.}$ If ${\tilde{E}}_{a,n}^r$ denotes the singly graded filtered-complex indexing, write $E_{a,b}^r={\tilde{E}}_{a,a+b}^r,d_r:E_{a,b}^r\to E_{a-r,b+r-1}^r\mathrm{.}$

The cohomological Serre filtration is the decreasing annihilator filtration $F^a{C}^n(E;R)=\ker (C^n(E;R)\to C^n(E_{a-1};R))\mathrm{.}$ Thus $F^0{C}^n=C^n$, and $\delta F^a{C}^n\subseteq F^a{C}^{n+1}$. Its reindexed differential convention is $d_r:E_r^{a,b}\to E_r^{a+r,b-r+1}\mathrm{.}$ These conventions include $E=\mathrm{\varnothing }$, the zero ring, $a=0$, and zero chains/cochains. The compact-support statement is applied separately to each specified simplex, so no choice principle is used.