Projective bundle formula is telling us that for a projective bundle $Y$ over $X$, where the fibers are $\mathbb{P}^n$, we have $G(Y)\cong G(X)^{n+1}$. Here $G$ is the $K$-theory of coherent sheaves. My question is that given an element in $G_n(Y)$ how can we recover the $n+1$ components of this as elements in $G(X)$? Let's consider the simplest case, where $Y=X\times \mathbb{P}^1$. By writing the localization sequence we have a short exact sequence $0\rightarrow G_n(X) \rightarrow G_n(X\times \mathbb{P}^1) \rightarrow G_n(X\times \mathbb{A}^1)\rightarrow 0$. The injection is given by pushforward along the inclusion of $\infty$, i.e. $\infty_*$. The surjection is given by pullback along inclusion. If we identify $G(X\times \mathbb{A}^1)$ with $G(X)$, it is basically pullback along the zero section i.e. $0^*$. Now this sequence splits (the surjection part splits by pullback along the projection $p:Y\rightarrow X$). So if we are given an element of $G_n(X\times \mathbb{P}^1)$, we can recover its second component by $0^*$. Now how can we recover the first component as an element of $G_n(X)$? Is there some morphism that pulling-back or pushing forward gives the first component?
Note the pull-back $\infty^*$ doesn't give the first component. Because of $\mathbb{A}^1$ invariance of $G$-theory it gives the same second component.
Perhaps the following will work. Observe that $K_0({\mathbb P}^m)$ acts on $G_n(X \times {\mathbb P}^m)$. Let $\zeta$ the class of the canonical line bundle in $K_0({\mathbb P}^m)$. Consider multiplication by $(1 - \zeta)^{m-i}$ followed by the push-forward map $p_*$. It seems to me that that ought to pick out the $i$-th component, where the components are numbered $0,\dots,m$. In the simple case you mentioned where $m=i=1$, the map is $p_*$.