无理代数群与 tame 分支

Wild ramification of nilpotent coverings and coverings of bounded degree

A finite étale map between irreducible, normal varieties is called tame, if it is tamely ramified with respect to all partial compactifications whose boundary is the support of a strict normal crossings divisor. We prove that if the Galois group of a Galois covering contains a normal nilpotent subgroup of index bounded by a constant N, then the covering is tame if and only if it is tamely ramified with respect to a single distinguished partial compactification only depending on N. The main tools used in the proof are Temkin's local purely inseparable uniformization and a Lefschetz type theorem…

Existence of slices in a tame context

We study the ramification theory for actions involving group schemes, focusing on the tame ramification. We consider the notion of tame quotient stack introduced in [AOV] and the one of tame action introduced in [CEPT]. We establish a local Slice theorem for unramified actions and after proving some interesting lifting properties for linearly reductive group schemes, we establish a Slice theorem for actions by commutative group schemes inducing tame quotient stacks. Roughly speaking, we show that these actions are induced from an action of an extension of the inertia group on a finitely presen…

Topology and Geometry of the Berkovich Ramification Locus for Rational Functions, II

This article is the second installment in a series on the Berkovich ramification locus for nonconstant rational functions f: P^1 -> P^1. Here we show the ramification locus of f is contained in a strong tubular neighborhood of finite radius around the connected hull of the critical points if and only if f is tamely ramified at all of its critical points. When the ground field has characteristic zero, this bound may be chosen to depend only on the residue characteristic. We give two applications to classical non-Archimedean analysis, including a new version of Rolle's theorem for rational funct…

On the ramification of étale cohomology groups

Let $K$ be a complete discrete valuation field whose residue field is perfect and of positive characteristic, let $X$ be a connected, proper scheme over $\mathcal{O}_K$, and let $U$ be the complement in $X$ of a divisor with simple normal crossings. Assume that the pair $(X,U)$ is strictly semi-stable over $\mathcal{O}_K$ of relative dimension one and $K$ is of equal characteristic. We prove that, for any smooth $\ell$-adic sheaf $\mathscr{G}$ on $U$ of rank one, at most tamely ramified on the generic fiber, if the ramification of $\mathscr{G}$ is bounded by $t+$ for the logarithmic upper rami…

On the ramified class field theory of relative curves

We generalize Deligne's approach to tame geometric class field theory to the case of a relative curve, with arbitrary ramification.

正则概形的驯同伦群的同伦不变性

Homotopy invariance of tame homotopy groups of regular schemes

针对Artin-Mazur平展同伦群仅特征零具同伦不变性的问题，本文引入禁野分歧的驯拓扑，证明正则概形的驯同伦群满足$\text{A}^1$-同伦不变性。

Generalization of Abhyankar's Lemma to henselian valued fields

Abhyankar showed that for a finite tame extension $L_1/K$ and a finite extension $L_2/K$ of $\mathfrak{P}$-adic fields, the condition $[νL_1 : νK]$ divides $[νL_2 : νK]$ is sufficient to eliminate ramification, that is, $L_1 \cdot L_2 / L_2$ is unramified. In this paper, we show that the above condition is not sufficient in the case of an arbitrary henselian valued field. We construct a counterexample illustrating that fact. We also give a necessary and sufficient condition for the elimination of tame ramification of a henselian field after a finite extension of the base field.

Forms of an affinoid disc and ramification

Let $k$ be a complete nonarchimedean field and let $X$ be an affinoid closed disc over $k$. We classify the tamely ramified twisted forms of $X$. Generalizing work of P. Russell on inseparable forms of the affine line we construct explicit families of wildly ramified forms of $X$. We compute the class group and the Grothendieck group of forms of $X$ in certain cases.

Ramification of surfaces: sufficient jet order for wild jumps

We develop a new approach to construction of numerical invariants for ramified coverings of algebraic surfaces of prime characteristic. Let A be a two-dimensional regular local ring of prime characteristic p with algebraically closed residue field. Let L/K be a solvable finite Galois extension of its fraction field. Let C be a germ of a regular curve on the spectrum of A which is not a component of the ramification divisor R. Then L/K determines (up to conjugation) an extension of the function filed of C. This is an extension of discrete valuation fields with algebaically closed residue fields…

非阿基米德曲线的三角剖分、半稳定约化与分歧

Triangulations of non-archimedean curves, semi-stable reduction, and ramification

本文利用 Berkovich 曲线的最小三角剖分研究非阿基米德曲线的半稳定约化，证明最小三角剖分重数的最小公倍数整除 Galois 扩张次数，并在驯顺情形下给出 Saito 定理的新证明及椭圆曲线和野分歧情形的推广分析。