# K/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 $$C \simeq D \times_X T$$. 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.

• What is "the $K$ Book"? – LSpice Oct 29 '20 at 3:07
• letmegooglethat.com/?q=the+K+book – Niels Oct 29 '20 at 7:27
• The "K Book" is Charles Weibel's Intro book on K Theory. – Leo Herr Oct 30 '20 at 3:11

The proof of Theorem 5.7 (2) of Hoyois and Krishna - Vanishing theorems for the negative $$K$$-theory of stacks shows that pullback is a homotopy equivalence of spectra $$G^{\text{naive}}(D) \overset{\sim}{\to} G^{\text{naive}}(C)$$ for any torsor $$C \to D$$ for a vector bundle $$E \to D$$. The hypotheses that the derived category is compactly generated and that $$D$$ is regular ensure the naive theory $$G^{\text{naive}}$$ coincides with Thomason–Trobaugh's $$G$$ Theory and $$K$$ Theory, respectively.
The proof uses an exact sequence $$0 \to E \to W \to \mathbb{A}^1_D \to 0$$ where $$W \times_{\mathbb{A}^1_D} (\{1\} \times D) \simeq C$$. This in turn gives a projective bundle $$\mathbb{P}(W)$$ with infinity section $$\mathbb{P}(E)$$ and open complement $$C$$. The localization sequence and the projective bundle formula of Krishna and Ravi - Algebraic $$K$$-theory of quotient stacks concludes.