Cohomology of a finite CW complex vanishes above its dimension

By [F1] and [F2] the singular cohomology $H^{\ast }(X,A;R)$ is the cohomology of the cellular cochain complex $C_{\mathrm{c}\mathrm{e}\mathrm{l}\mathrm{l}}^{\ast }(X,A;R)$ built from the relative cells of the skeletal filtration.

A finite CW complex of dimension $d$ has $X^n=X^{n-1}=X$ for $n>d$. The degree-$n$ cellular cochain group from the skeletal filtration of [F1] is the relative cohomology of the consecutive skeleta $(X^n\cup A,X^{n-1}\cup A)$, which is therefore the zero group. Thus $C_{\mathrm{c}\mathrm{e}\mathrm{l}\mathrm{l}}^n(X,A;R)=0$ for $n>d$.

A cochain complex whose groups vanish in all degrees above $d$ has cohomology zero above $d$, since a degree-$n$ cohomology class for $n>d$ is a class in a zero group; therefore $H^n(X,A;R)=0$ for $n>d$.

In the absolute case $A=\mathrm{\varnothing }$ this gives $H^n(X;R)=0$ for $n>d$, which is the first assertion.

Boundary cases. For $d=0$ the complex is a finite discrete set, all cochains vanish in positive degrees and the statement reads $H^n(X;R)=0$ for $n\ge 1$, which holds because $X$ is a disjoint union of points. For the empty complex both sides vanish in every degree (the empty CW complex has dimension $-\mathrm{\infty }$ by convention, and the statement is vacuous). Negative degrees are outside the assertion. The ring $R$ may be the zero ring, in which case all groups vanish; no division or flatness is used. AC is used only through [A1] in the cellular comparison.