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| Formato: | Preprint |
| Publicado: |
2018
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| Acceso en línea: | https://arxiv.org/abs/1807.06400 |
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| _version_ | 1866913225226846208 |
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| author | Deninger, Christopher |
| author_facet | Deninger, Christopher |
| contents | Motivated by work of Kucharczyk and Scholze, we use sheafified rational Witt vectors to attach a new ringed space $W_{\mathrm{rat}} (X)$ to every scheme $X$. We also define $R$-valued points $W_{\mathrm{rat}} (X) (R)$ of $W_{\mathrm{rat}} (X)$ for every commutative ring $R$. For normal schemes $X$ of finite type over spec $\mathbb{Z}$, using $W_{\mathrm{rat}} (X) (\mathbb{C})$ we construct infinite dimensional $\mathbb{R}$-dynamical systems whose periodic orbits are related to the closed points of $X$. Various aspects of these topological dynamical systems are studied. We also explain how certain $p$-adic points of $W_{\mathrm{rat}} (X)$ for $X$ the spectrum of a $p$-adic local number ring are related to the points of the Fargues-Fontaine curve. The new intrinsic construction of the dynamical systems generalizes and clarifies the original extrinsic construction in v.1 and v.2. Many further results have been added. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_1807_06400 |
| institution | arXiv |
| publishDate | 2018 |
| record_format | arxiv |
| spellingShingle | Dynamical systems for arithmetic schemes Deninger, Christopher Dynamical Systems Algebraic Geometry Motivated by work of Kucharczyk and Scholze, we use sheafified rational Witt vectors to attach a new ringed space $W_{\mathrm{rat}} (X)$ to every scheme $X$. We also define $R$-valued points $W_{\mathrm{rat}} (X) (R)$ of $W_{\mathrm{rat}} (X)$ for every commutative ring $R$. For normal schemes $X$ of finite type over spec $\mathbb{Z}$, using $W_{\mathrm{rat}} (X) (\mathbb{C})$ we construct infinite dimensional $\mathbb{R}$-dynamical systems whose periodic orbits are related to the closed points of $X$. Various aspects of these topological dynamical systems are studied. We also explain how certain $p$-adic points of $W_{\mathrm{rat}} (X)$ for $X$ the spectrum of a $p$-adic local number ring are related to the points of the Fargues-Fontaine curve. The new intrinsic construction of the dynamical systems generalizes and clarifies the original extrinsic construction in v.1 and v.2. Many further results have been added. |
| title | Dynamical systems for arithmetic schemes |
| topic | Dynamical Systems Algebraic Geometry |
| url | https://arxiv.org/abs/1807.06400 |