$K_1(Var)$ is presented by stratified birational equivalences

TL;DR

This paper fully describes the $K_1$ group of varieties using stratified birational equivalences, extending $K$-theory to broader categories and highlighting new automorphism concepts.

Contribution

It adapts Gillet-Grayson's $G$-Construction to define an un-delooped $K$-theory spectrum for varieties and introduces stratified birational equivalences as generators of $K_1$.

Findings

Streamlined and extended $K$-theory results for categories including $Var$.

Defined stratified birational equivalences as generators of $K_1$.

Extended the construction to non-additive contexts like o-minimal structures.

Abstract

This paper provides a complete presentation of $K_1(Var)$, the $K_1$ group of varieties, resolving and simplifying a problem left open in \cite{ZakhK1}. Our approach adapts Gillet-Grayson's $G$-Construction to define an un-delooped $K$-theory spectrum of varieties. There are two levels on which one can read the present paper. On a technical level, we streamline and extend previous results on the $K$-theory of exact categories to a broader class of categories, including $Var$. On a more conceptual level, our investigations bring into focus an interesting generalisation of automorphisms (``double exact squares'') which generate $K_1$. For varieties, this corresponds to what we call stratified birational equivalences, but the construction extends to a wide range of non-additive contexts (e.g. $o$-minimal structures, definable sets etc.). This raises a challenging question: what kind of information do these generalised automorphisms calibrate?