Algebraic K-theory and cubical descent

TL;DR

This paper introduces a descent variant of algebraic K-theory for varieties over characteristic zero fields, establishing its equivalence to homotopy K-theory and revealing a natural weight filtration on its groups.

Contribution

It defines a new descent-based algebraic K-theory for varieties, connecting it with existing homotopy K-theory and analyzing its structural properties.

Findings

The descent variant coincides with classical K-theory for smooth varieties.

The new theory is equivalent to Weibel's homotopy algebraic K-theory.

A natural weight filtration on homotopy K-groups is established.

Abstract

In this note we apply Guillen-Navarro descent theorem, \cite{GN02}, to define a descent variant of the algebraic $K$-theory of varieties over a field of characteristic zero, $\mathcal{KD}(X)$, which coincides with $\mathcal{K}(X)$ for smooth varieties. After a result of Haesemeyer, this new theory is equivalent to the homotopy algebraic $K$-theory introduced by Weibel. We also prove that there is a natural weight filtration on the groups $KH_\ast(X)$.