Weight Filtrations and Derived Motivic Measures
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arXiv
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| Format: | Preprint |
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2024
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| _version_ | 1866913194301194240 |
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| author | Nanavaty, Anubhav |
| author_facet | Nanavaty, Anubhav |
| contents | Let $k$ be a field admitting resolution of singularities. We lift a number of motivic measures, such as the Gillet-Soulé measure and the compactly supported $\mathbb{A}^1$-Euler characteristic, to derived motivic measures in the sense of Campbell-Wolfson-Zakharevich, answering various questions in the literature. We do so by generalizing the construction of the Gillet-Soulé weight complex to show that it is well-defined up to a certain notion of weak equivalence in the category of simplicial smooth projective varieties. For a $k$-variety $X$, the collection of all Gillet-Soulé weight complexes of $X$ form a 'weakly constant' pro-object of simplicial varieties, and under mild assumptions, the $K$-theory of a Waldhausen category is equivalent to the $K$-theory of its weakly constant pro-objects. This leads us to a new proof of the existence of the Gillet-Soulé weight filtration, along with the weight filtration on both the stable and unstable homotopy type of a variety over $k$. We show these constructions provide the aforementioned derived motivic measures, or maps of spectra, out of $K(Var_k)$, the Zakharevich $K$-theory of varieties. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2401_06879 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Weight Filtrations and Derived Motivic Measures Nanavaty, Anubhav Algebraic Geometry Algebraic Topology Let $k$ be a field admitting resolution of singularities. We lift a number of motivic measures, such as the Gillet-Soulé measure and the compactly supported $\mathbb{A}^1$-Euler characteristic, to derived motivic measures in the sense of Campbell-Wolfson-Zakharevich, answering various questions in the literature. We do so by generalizing the construction of the Gillet-Soulé weight complex to show that it is well-defined up to a certain notion of weak equivalence in the category of simplicial smooth projective varieties. For a $k$-variety $X$, the collection of all Gillet-Soulé weight complexes of $X$ form a 'weakly constant' pro-object of simplicial varieties, and under mild assumptions, the $K$-theory of a Waldhausen category is equivalent to the $K$-theory of its weakly constant pro-objects. This leads us to a new proof of the existence of the Gillet-Soulé weight filtration, along with the weight filtration on both the stable and unstable homotopy type of a variety over $k$. We show these constructions provide the aforementioned derived motivic measures, or maps of spectra, out of $K(Var_k)$, the Zakharevich $K$-theory of varieties. |
| title | Weight Filtrations and Derived Motivic Measures |
| topic | Algebraic Geometry Algebraic Topology |
| url | https://arxiv.org/abs/2401.06879 |