Categorical Milnor squares and K-theory of algebraic stacks

TL;DR

This paper develops a new framework for understanding algebraic K-theory of algebraic stacks using Milnor squares of stable $ abla$-categories, proving excision theorems and generalizing Weibel's conjecture.

Contribution

It introduces a notion of Milnor squares for stable $ abla$-categories and applies it to establish K-theoretic excision and vanishing results for algebraic stacks.

Findings

Milnor squares induce cartesian squares in K-theory under certain conditions

Proves Milnor and proper excision theorems for algebraic stacks with affine diagonal

Generalizes Weibel's conjecture on negative K-groups for these stacks

Abstract

We introduce a notion of Milnor square of stable $\infty$-categories and prove a criterion under which algebraic K-theory sends such a square to a cartesian square of spectra. We apply this to prove Milnor excision and proper excision theorems in the K-theory of algebraic stacks with affine diagonal and nice stabilizers. This yields a generalization of Weibel's conjecture on the vanishing of negative K-groups for this class of stacks.