Ramification filtration via deformations,Sbornik: Mathematics

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Ramification filtration via deformations
Sbornik: Mathematics ( IF 0.8 ) Pub Date : 2021-04-14 , DOI: 10.1070/sm9322
V. A. Abrashkin _{1,

2}

Affiliation

Let $\mathscr K$ be a field of formal Laurent series with coefficients in a finite field of characteristic $p$ , $\mathscr G_{<p}$ the maximal quotient of the Galois group of $\mathscr K$ of period $p$ and nilpotency class $<p$ and $\{\mathscr G_{<p}^{(v)}\}_{v\geqslant 1}$ the filtration by ramification subgroups in the upper numbering. Let $\mathscr G_{<p}=G(\mathscr L)$ be the identification of nilpotent Artin-Schreier theory: here $G(\mathscr L)$ is the group obtained from a suitable profinite Lie $\mathbb{F}_p$ -algebra $\mathscr L$ via the Campbell-Hausdorff composition law. We develop a new technique for describing the ideals $\mathscr L^{(v)}$ such that $G(\mathscr L^{(v)})=\mathscr G_{<p}^{(v)}$ and constructing their generators explicitly. Given $v_0\geqslant 1$ , we construct an epimorphism of Lie algebras $\overline\eta^{\unicode{8224}}\colon \mathscr L\to \overline{\mathscr L}^{\unicode{8224}}$ and an action $\Omega_U$ of the formal group of order $p$ , $\alpha_p=\operatorname{Spec}\mathbb{F}_p[U]$ , $U^p=0$ , on $\overline{\mathscr L}^{\unicode{8224}}$ . Suppose $d\Omega_U=B^{\unicode{8224}}U$ , where $B^{\unicode{8224}}\in\operatorname{Diff}\overline{\mathscr L}^{\unicode{8224}}$ , and $\overline{\mathscr L}^{\unicode{8224}}[v_0]$ is the ideal of $\overline{\mathscr L}^{\unicode{8224}}$ generated by the elements of $B^{\unicode{8224}}(\overline{\mathscr L}^{\unicode{8224}})$ . The main result in the paper states that $\mathscr L^{(v_0)}=(\overline\eta^{\unicode{8224}})^{-1}\overline{\mathscr L}^{\unicode{8224}}[v_0]$ . In the last sections we relate this result to the explicit construction of generators of $\mathscr L^{(v_0)}$ obtained previously by the author, develop a more efficient version of it and apply it to recover the whole ramification filtration of $\mathscr G_{<p}$ from the set of its jumps.

Bibliography: 13 titles.

中文翻译：

通过变形进行分枝过滤

令 $\mathscr K$ 是一个正式的 Laurent 级数域，其系数在特征的有限域中， $p$ 周期和幂等 $\mathscr G_{<p}$ 性的伽罗瓦群的最大商以及上部编号中的分支子群的过滤。设幂零 Artin-Schreier 理论的识别：这是通过 Campbell-Hausdorff 组合定律从合适的有限李代数获得的群。我们开发了一种新技术来描述理想并明确地构建它们的生成器。给定, 我们构造李代数的同态和形式阶群的作用. $\mathscr K$ $p$ $<p$ $\{\mathscr G_{<p}^{(v)}\}_{v\geqslant 1}$ $\mathscr G_{<p}=G(\mathscr L)$ $G(\mathscr L)$ $\mathbb{F}_p$ $\mathscr L$ $\mathscr L^{(v)}$ $G(\mathscr L^{(v)})=\mathscr G_{<p}^{(v)}$ $v_0\geqslant 1$ $\overline\eta^{\unicode{8224}}\冒号\mathscr L\to \overline{\mathscr L}^{\unicode{8224}}$ $\Omega_U$ $p$ , $\alpha_p=\operatorname{规范}\mathbb{F}_p[U]$ , $U^p=0$ , 上 $\overline{\mathscr L}^{\unicode{8224}}$ 。假设 $d\Omega_U=B^{\unicode{8224}}U$ , 其中 $B^{\unicode{8224}}\in\operatorname{Diff}\overline{\mathscr L}^{\unicode{8224}}$ 和 $\overline{\mathscr L}^{\unicode{8224}}[v_0]$ 是的 $\overline{\mathscr L}^{\unicode{8224}}$ 元素生成的理想值 $B^{\unicode{8224}}(\overline{\mathscr L}^{\unicode{8224}})$ 。论文的主要结果表明 $\mathscr L^{(v_0)}=(\overline\eta^{\unicode{8224}})^{-1}\overline{\mathscr L}^{\unicode{8224}}[v_0]$ 。在最后几节中，我们将这个结果与作者之前获得的生成器的显式构造联系起来，开发了一个更有效的版本，并将其应用于从其跳跃集合中 $\mathscr L^{(v_0)}$ 恢复整个分支过滤。 $\mathscr G_{<p}$

参考书目：13 个标题。

更新日期：2021-04-14

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