$f : C \to D$ is a morphism of artin stacks over $X$ which is a torsor for a vector bundle $T \to X$: etale-locally in $X$, we have $C \simeq D \times_X T$. I want to conclude that $f^*: G(D) \to G(C)$ (or the same for $K$ Theory) is injective.

Exercise V 6.6 in the $K$ Book states that for flat maps $E \to X$ of noetherian schemes with fibers given by affine space, the pullback is an equivalence on $G$ Theory. The fibers over geometric points of $f$ are affine space, but it could be that many fibers have no sections. Moreover, I want to use this fact on Artin stacks instead of schemes, defining their $G$ Theory/$K$ Theory using sites introduced by Olsson in "Sheaves on Artin Stacks" as elaborated in Feng Qu's "Virtual Pullbacks in $K$ Theory."

Thanks for the help -- I'm somewhat new to $K$ Theory.

Setting: $f : C \to D$ is a morphism of Artin stacks over $X$ which is a torsor for a vector bundle $T \to X$: étale-locally in $X$, we have $C \simeq D \times_X T$. I want to conclude that $f^*: G(D) \to G(C)$ (or the same for $K$-theory) is injective.

Exercise V 6.6 in the $K$-book by Ch. Weibel states that for flat maps $E \to X$ of noetherian schemes with fibers given by affine space, the pullback is an equivalence on G-theory. The fibers over geometric points of $f$ are affine space, but it could be that many fibers have no sections. Moreover, I want to use this fact on Artin stacks instead of schemes, defining their G-theory/K-theory using sites introduced by Olsson in "Sheaves on Artin stacks" as elaborated in Feng Qu's "Virtual pullbacks in K-theory."

Thanks for the help — I'm somewhat new to K-theory.