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ag.algebraic geometry – Ok/G-theory of affine bundles

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 now $C simeq D times_X T$. I need to resolve that $f^*: G(D) to G(C)$ (or the identical for $Ok$-theory) is injective.

Exercise V 6.6 within the $Ok$-book by Ch. Weibel states that for flat maps $E to X$ of noetherian schemes with fibers given by affine house, the pullback is an equivalence on G-theory. The fibers over geometric factors of $f$ are affine house, however it might breathe that many fibers don’t have any sections. Moreover, I need to employ this reality on Artin stacks as an alternative of schemes, defining their G-theory/Ok-theory utilizing websites launched by Olsson in *“Sheaves on Artin stacks”* as elaborated in Feng Qu’s *“Virtual pullbacks in K-theory.”*

Thanks for the ameliorate — I’m considerably fresh to Ok-theory.

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