I am having trouble with computing the cohomology ring of the flag bundle of $V \to X$. It seems that I should consider the flag bundle as an iterated projective bundle and utilize the defining feature of the chern classes (namely that they give the coefficients of the relation in the projective bundle), but beyond this outline I'm rather confused. Is there a textbook or other source that explains this? I have looked through both Hatcher texts and also online without much success.
If there are any questions about what I am asking please comment.
Question: "I have looked through both Hatcher texts and also online without much success. If there are any questions about what I am asking please comment."
Answer: There is a formula "- the projective bundle formula" - which holds for a large class of cohomology theories. In the case of a locally trivial sheaf $E$ of rank $n+1$ on an algebraic variety $X$ it calculates the "cohomology group" (the Chow group) of the associated projective bundle:
$$PB.\text{ }CH^*(\mathbb{P}(E^*)) \cong CH^*(X)[t]/(t^{n+1}).$$
The formula says that $CH^*(\mathbb{P}(E^*))$ is a free $CH^*(X)$-module on the elements
$$ \{1,\overline{t},\ldots ,\overline{t^n} \}.$$
The formula $PE$ may be used to define the Chern classes of $E$.
The "full flag bundle" $Fl(E)$ of $E$ may be constructed as a sequence of projective bundles, hence the formula $PB$ may be used to calculate the Chow group $CH^*(Fl(E))$.
A similar approach may be used for the calculation of the cohomology $H^*(Fl(E))$ where $\pi:Fl(E) \rightarrow X$ is a flag bundle and $X$ is a complex manifold or a real differentiable manifold - search for "projective bundle formula". Similar formulas hold for the grothendieck group.