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I have a nontrivial gerbe $\pi : \mathscr{G} \to X$ banded by a cyclic group $G = \mathbb{Z}/r$. I'm working over $\mathbb{C}$. I want to describe $\pi_\ast$ and relate the fundamental class $[\mathcal{O}_X]$ with the pushforward of the fundamental class $[\mathcal{O}_{\mathscr{G}}]$ in $K$ theory. If it helps, $X$ is a smooth algebraic stack.

In chow with $\mathbb{Q}$ coefficients, one has $\pi_\ast [\mathscr{G}] = \dfrac{1}{r} [X]$ because it is a morphism proper of pure degree $\dfrac{1}{r}$. If $\mathscr{G} = BG_X$ is a trivial gerbe over $X$, the sequence $X \overset{p}{\to} BG_X \overset{\pi}{\to} X$ and the formula for pushforward for torsors $p$ gives $$p_\ast[\mathcal{O}_{X}] = [\mathcal{O}_X^{\oplus \#G}],$$ where the latter sheaf has $G$ action permuting the coordinates (if I have this right). I'm not sure how to compare this with $[\mathcal{O}_{BG_X}]$, which I believe is $[\mathcal{O}_X]$ with the natural $G$ action. I am then told that $\pi_\ast$ takes $G$-invariant sections.

I can live with inverting a few numbers, $K() \otimes \mathbb{Z}[\dfrac{1}{r_1 r_2 \dots, r_k}]$ if necessary to get a relationship. I don't want to just tensor with $\mathbb{Q}$ because then $K$ theory is basically just chow.

Questions:

  1. What is the pushforward $[\mathcal{O}_{\mathscr{G}}]$?
  2. How is it related to $[\mathcal{O}_X]$? Is there a natural variant of the fundamental class of $\mathscr{G}$ that pushes forward precisely to that of $X$? Something like $r[\mathscr{G}]$ in chow?
  3. Do I have the above right? What does this look like in the example of a trivial gerbe?
  4. References for $K$ theory on gerbes would be greatly appreciated.
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