Vanishing theorems for the negative $K$-theory of stacks

Abstract

We prove that the homotopy algebraic $K$-theory of tame quasi-DM stacks satisfies cdh-descent. We apply this descent result to prove that if $\mathcal{X}$ is a Noetherian tame quasi-DM stack and $i<-dim\left(\mathcal{X}\right)$, then $K_i\left(\mathcal{X}\right)\left[1∕n\right]=0$ if $n$ is nilpotent on $\mathcal{X}$ and $K_i\left(\mathcal{X},\mathbb{Z}∕n\right)=0$ if $n$ is invertible on $\mathcal{X}$. Our descent and vanishing results apply more generally to certain Artin stacks whose stabilizers are extensions of finite group schemes by group schemes of multiplicative type.