$K_1(Var)$ is presented by stratified birational equivalences
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arXiv
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| Format: | Preprint |
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2025
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| _version_ | 1866913006361772032 |
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| author | Ng, Ming |
| author_facet | Ng, Ming |
| contents | 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? |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2510_20433 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | $K_1(Var)$ is presented by stratified birational equivalences Ng, Ming K-Theory and Homology Algebraic Geometry Algebraic Topology 19D99, 13D15, 14E07, 19E99, 18F30 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? |
| title | $K_1(Var)$ is presented by stratified birational equivalences |
| topic | K-Theory and Homology Algebraic Geometry Algebraic Topology 19D99, 13D15, 14E07, 19E99, 18F30 |
| url | https://arxiv.org/abs/2510.20433 |