Given $X$, and a locally free sheaf $\mathcal{F}$ of rank $n$ on it. We have the induced map $f:\mathbb{P}(\mathcal{F})\rightarrow X$. Let $\phi$ be the divisor associated to the line bundle $\mathcal{O}_{\mathbb{P}(\mathcal{F})}(1) $.
Is it true $f_{\ast}(\phi^{n-1})=[X]$ in the Chow ring $A^\ast(X)$? Also, is trace map $A^{top}(.)\rightarrow\mathbb{Q}$ compatible with pushforward $f_{\ast}$?
Thank you.
The class $\phi^{n-1}$ has degree $1$ when restricted to each fibre of $f$ since $H^{n-1}$ has degree $1$ on $\mathbb{P}^{n-1}$, where $H$ is the class of a hyperplane, i.e. a divisor associated to $\mathcal{O}(1)$. It follows that $f_*(\phi^{n-1})$ is indeed $[X]$ in $A^*(X)$.
I suppose what you call the trace map is just the degree map $A^{top}(X) \to \mathbb{Z}$ (for a proper variety $X$). It follows immediately from the definition of proper pushforward that it is compatible with $f_*$.
