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arXiv · 2510.20433

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

Ming Ng
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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?

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Ming Ng. 2025-10-23. $K_1(Var)$ is presented by stratified birational equivalences. https://arxiv.org/abs/2510.20433

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