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.

  • 2
    $\begingroup$ What is "the $K$ Book"? $\endgroup$ – LSpice Oct 29 '20 at 3:07
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    $\begingroup$ letmegooglethat.com/?q=the+K+book $\endgroup$ – Niels Oct 29 '20 at 7:27
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    $\begingroup$ The "K Book" is Charles Weibel's Intro book on K Theory. $\endgroup$ – 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.

Thanks to Adeel Khan for pointing me to these references.

EDIT: Adeel Khan just wrote a paper with a more direct reference. See Theorem 3.16 of K-theory and G-theory of derived algebraic stacks.


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