Cohomology of projective bundle only depends on base and fiber?

Question

Let $P\to X$ be a $\mathbb P^n$ bundle. Is it true that all the (co)homology group only depends on $X$ and $n$ (and independent of the transform funcions) ?

Answer 1

This is true and this is essentially the content of the projective bundle theorem. You can find it stated here : https://en.wikipedia.org/wiki/Projective_bundle

But note however that even if the cohomology groups $H^*(P)$ only depends on the base scheme $X$, this is not true for the ring structure. In fact we have : $$ H^*(P)=H^*(X)[\xi]/(\xi^n+c_1\xi^{n-1}+...+c_{n+1}\xi^0)$$ where $\xi\in H^2(P)$ is of degree 2 and is the class $c_1(\mathcal{O}_{P}(1))$ of the tautological bundle, and $c_1,...,c_{n+1}\in H^*(X)$ are the Chern classes of $P$.