Take a ring $R$ and define $cd_R(X)$ ("cohomological dimension over $R$") to be the maximum of the integers $m$ such that $H^m(X; R)\neq 0$, where $H^*$ denotes the Alexander-Spanier cohomology.
Now let's say that I have a compact Lie group $G$ which acts on a compact Hausdorff space $X$. Is it true that $cd_R(X/G) \leq cd_R(X)$?
I am aware of Quillen's result which tackles this sort of question for sheaf cohomology. I know that sheaf cohomology coincides with Alexander-Spanier cohomology for compact spaces, so I thought this is it, but then I realized that definition of cohomological dimension in Quillen's paper is broader. Namely, he takes the supremum over the integers $m$ such that there exists a sheaf $F$ of $R$-modules on $X$ with $H^m(X; F) \neq 0$ (i.e. the supremum is also taken over all possible sheaves, whereas I want to compare cohomology groups for fixed coefficients).
So perhaps it is possible that there exists a particularly nasty ring $R$ such that $H^m(X/G; R)\neq 0$, but $H^i(X;R)=0$ for $i\geq m$?
Sorry if this is a silly question. I know next to nothing about sheaves, but will learn some if Quillen's theorem actually deals with my question. :-)