Chow group of flag variety

Question

Let $X$ be a complex variety and $E$ be a vector bundle over $X$ of rank $n$. Let $F$ be the associated complete flag manifold bundle over $X$, i.e., associated to each point $x\in X$, the fibre $F_x$ is the complete flag manifold of $E_x$. Then how to compute the Chow group of it? The hint said that factorize $F\rightarrow X$ through $\mathbb{P}(E^*)$, but I cannot see such a factorization.

Answer 1

Question: "The hint said that factorize $F→X$ through $P(E^∗)$, but I cannot see such a factorization."

Answer: If $E:=E_1$ is a rank $e+1$ locally trivial $\mathcal{O}_X$-module, define $\pi_1: X_1:=\mathbb{P}(E_1^*)\rightarrow X$. You get an exact sequence of locally free sheaves

$$0 \rightarrow \mathcal{O}(-1) \rightarrow \pi_1^*E_1 \rightarrow E_2 \rightarrow 0$$ with $rk(E_2)=rk(E)-1$. Define simlarly $\pi_2: X_2:=\mathbb{P}(E_2^*)\rightarrow X_1$ and (by induction) you get a sequence

$$X_e \rightarrow X_{e-1} \rightarrow \cdots X_2 \rightarrow X_1 \rightarrow X$$

where each map in the sequence is a projective bundle. For each map you can use the "projective bundle formula":

$$CH^*(\mathbb{P}(E_i^*)) \cong CH^*(\mathbb{P}(E_{i-1}^*))[t]/(t^{e_i})$$

where $rk(E_i)=e_i$. By induction you get a formula for $CH^*(\mathbb{P}(E_e^*))$ in terms of $CH^*(X)$. You find an explanation in Fulton (Intersection Theory) Example 3.3.5. The scheme $F:=\mathbb{P}(E_e^*)$ is the "complete flag bundle" of $E$.