# The $K$-theory of toric schemes over regular rings of mixed characteristic

## TL;DR

This paper proves homotopy invariance of the direct limit of K-groups for toric schemes over regular rings of mixed characteristic, extending known results to more general base rings and algebra types.

## Contribution

It generalizes the homotopy invariance of K-groups for toric schemes from rings containing a field to regular rings of mixed characteristic and to certain non-commutative K-regular algebras.

## Key findings

- Homotopy invariance holds for toric schemes over regular rings of mixed characteristic.
- The affine case was conjectured by Gubeladze and is proven here.
- Results extend to non-commutative K-regular algebras.

## Abstract

We show that if $X$ is a toric scheme over a regular commutative ring $k$ then the direct limit of the $K$-groups of $X$ taken over any infinite sequence of nontrivial dilations is homotopy invariant. This theorem was previously known for regular commutative rings containing a field. The affine case of our result was conjectured by Gubeladze. We prove analogous results when $k$ is replaced by an appropriate $K$-regular, not necessarily commutative $k$-algebra.