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

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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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