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| Format: | Preprint |
| Published: |
2022
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| Subjects: | |
| Online Access: | https://arxiv.org/abs/2206.11688 |
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| _version_ | 1866910658388295680 |
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| author | Lerbet, Samuel |
| author_facet | Lerbet, Samuel |
| contents | We relate the group structure of van der Kallen on orbit sets of unimodular rows with values in a smooth algebra $A$ over a field $k$ with the motivic cohomotopy groups of the spectrum of $A$ with coefficients in $\mathbb{A}^n\setminus 0$ in the sense of Asok and Fasel. In the last section, we compare the motivic cohomotopy theory studied in this paper and defined by $\mathbb{A}^{n+1}\setminus 0$ or, equivalently, by an $\mathbb{A}^1$-weakly equivalent quadric $Q_{2n+1}$ to that considered by Asok and Fasel, defined by a quadric $Q_{2n}$, by means of explicit morphisms $Q_{2n+1}\rightarrow Q_{2n}$, $Q_{2n}\times\mathbb{G}_m\rightarrow Q_{2n+1}$ of quadrics. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2206_11688 |
| institution | arXiv |
| publishDate | 2022 |
| record_format | arxiv |
| spellingShingle | Motivic stable cohomotopy and unimodular rows Lerbet, Samuel K-Theory and Homology Algebraic Geometry We relate the group structure of van der Kallen on orbit sets of unimodular rows with values in a smooth algebra $A$ over a field $k$ with the motivic cohomotopy groups of the spectrum of $A$ with coefficients in $\mathbb{A}^n\setminus 0$ in the sense of Asok and Fasel. In the last section, we compare the motivic cohomotopy theory studied in this paper and defined by $\mathbb{A}^{n+1}\setminus 0$ or, equivalently, by an $\mathbb{A}^1$-weakly equivalent quadric $Q_{2n+1}$ to that considered by Asok and Fasel, defined by a quadric $Q_{2n}$, by means of explicit morphisms $Q_{2n+1}\rightarrow Q_{2n}$, $Q_{2n}\times\mathbb{G}_m\rightarrow Q_{2n+1}$ of quadrics. |
| title | Motivic stable cohomotopy and unimodular rows |
| topic | K-Theory and Homology Algebraic Geometry |
| url | https://arxiv.org/abs/2206.11688 |