Let $E \to X$ be a vector bundle over an algebraic variety and $P(E)$ the projectivized vector bundle. In class we saw that $A^*(P(E)) = A^*(X)[\xi]/(\xi^r + c_1(E)\xi^{r-1} + \dots + c_r)$. $(\star)$
1) What are the hypothesis for this to be true ?
Assume it's true and let's consider $E = O(a) \oplus O(b) \oplus O(c)$ over $\mathbb P^l$. We have $c(E) = 1 + (a + b + c)h + (ab + bc + ac)h^2 + abc h^3$. According to $(\star)$ we shoud have $$A^*(P(E)) = \mathbb Z[h, \xi]/(h^l, \xi^3 + (a+b+c)\xi^2 h + (ab + bc + cd)\xi h^2 + abc h^3)$$
But on the other hand, we have $P(E) \cong P(E')$ where $E' = O(a-c) \oplus O(b-c) \oplus O$. This gives $$A^*(P(E)) = \mathbb Z[h, \xi]/(h^l, \xi^3 + (a + b - 2 c)\xi^2 h + (a-c)(b-c)\xi h^2)$$
These presentations look pretty different, so my question is :
2) Are they the same ? If not, where is my mistake ?