# Noncommutative rigidity

## TL;DR

This paper proves invariance properties of algebraic K-theory and Grothendieck groups under field extensions for dg categories, extending classical rigidity theorems to noncommutative and singular settings.

## Contribution

It introduces invariance results for noncommutative invariants under field extensions and extends classical rigidity theorems to singular varieties and noncommutative motives.

## Key findings

- Invariance of the numerical Grothendieck group under primary field extensions.
- Invariance of mod-n algebraic K-theory under separably closed field extensions.
- Introduction of the category of n-adic noncommutative mixed motives.

## Abstract

In this article we prove that the numerical Grothendieck group of every smooth proper dg category is invariant under primary field extensions, and also that the mod-n algebraic K-theory of every dg category is invariant under extensions of separably closed fields. As a byproduct, we obtain an extension of Suslin's rigidity theorem, as well as of Yagunov-Ostvaer's equivariant rigidity theorem, to singular varieties. Among other applications, we show that base-change along primary field extensions yields a faithfully flat morphism between noncommutative motivic Galois groups. Finally, along the way, we introduce the category of n-adic noncommutative mixed motives.