$\mathbb{A}^1$-invariance of localizing invariants

TL;DR

This paper extends Weibel's $A^1$-invariance results for K-theory to all finitary localizing invariants of small stable $Infinity$-categories, using categorical tools and endofunctors.

Contribution

It generalizes $A^1$-invariance from K-theory to all finitary localizing invariants of small stable $Infinity$-categories.

Findings

Frobenius and Verschiebung endofunctors are studied categorically.

A categorical version of Stienstra's projection formula is established.

The $A^1$-invariance property is extended to broader invariants.

Abstract

Weibel proved that $p$-inverted K-theory is $\mathbb{A}^1$-invariant on $\mathbb{F}_p$-schemes and K-theory with $\mathbb{Z}/p$-coefficients is $\mathbb{A}^1$-invariant on $\mathbb{Z}[\frac{1}{p}]$-schemes. We extend this result to all finitary localizing invariants of small stable $\infty$-categories. Along the way we study the Frobenius and Verschiebung endofunctors defined by Tabuada and provide a categorical version of Stienstra's projection formula.