# Vanishing theorems for the negative K-theory of stacks

## TL;DR

This paper proves cdh-descent for homotopy algebraic K-theory of tame quasi-DM stacks and establishes vanishing results for negative K-theory groups under certain conditions, advancing understanding of K-theory in algebraic stacks.

## Contribution

It introduces cdh-descent for homotopy K-theory of tame stacks and proves new vanishing theorems for negative K-groups of such stacks.

## Key findings

- Homotopy algebraic K-theory satisfies cdh-descent for tame quasi-DM stacks.
- Negative K-groups vanish below the negative dimension of the stack under certain conditions.
- Results extend to certain Artin stacks with specific stabilizer structures.

## 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 X is a Noetherian tame quasi-DM stack and i < -dim(X), then K_i(X)[1/n] = 0 (resp. K_i(X, Z/n) = 0) provided that n is nilpotent on X (resp. is invertible on 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.