One-dimensional fibers of rigid subanalytic sets
One-dimensional fibers of rigid subanalytic sets
复制标题
刚性亚解析集的一维纤维
DOI：
10.2307/2586589
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发表时间：
1998
影响因子：
0.6
通讯作者：
Z. Robinson
中科院分区：
文献类型：
--
作者：
L. Lipshitz;Z. Robinson
设K是一个具有任意特征的代数闭域，关于非平凡超度量绝对值λ·λ完备：K → λ +。R表示K的赋值环，R表示K的极大理想.我们在[5]中定义的次解析集类中工作，但我们的结果在这里也适用于[11]中引入的强次解析集以及[6]中考虑的那些次解析集。设X ∈ R1是亚解析的。在[8]中，我们证明了X的分解是有限个特殊集合U <$R1的并集（见下文）。在本文中，在定理1.6中，我们得到了这个结果的一个版本，它在参数上是一致的，从而回答了安格斯麦金太尔引起我们注意的一个问题。从定理1.6可以直接得出，在[3]和[9]的意义下，语言中的K理论（见[5]和[6]）是C-极小的。最近在[12]中证明了在p-adic情况下的类似一致性结果。定义1.1. (i)R1中的圆盘是以下两种形式之一的集合：R1中的特殊集合是圆盘减去圆盘的有限并集。(ii)R-域u ∈ Rm和它们的解析函数环，归纳定义如下。R_m是R-整环，是K上X_1，…，X_m中严格收敛的幂级数环。如果u是一个有关联环的R-整环，（其中K <$X，Y <$$> ρ <$S是分离幂级数的环，参见[5，§2]和[1，§1]），f在u上没有公共零点，并且f {<，≤}，则是R-域，其中J是由I和f − gZ生成的理想（Z是一个新变量），如果≤，并且其中J是由I和f − gτ（τ是一个新变量）生成的理想，如果<。(See[8，定义2.2]。R-整环推广了[2，§7.2.3]中的有理整环。这是真的，但不容易证明，只依赖于u作为一个点集，是独立的特定表示的u。
Let K be an algebraically closed field of any characteristic, complete with respect to the non-trivial ultrametric absolute value ∣·∣: K → ℝ+. By R denote the valuation ring of K, and by ℘ its maximal ideal. We work within the class of subanalytic sets defined in [5], but our results here also hold for the strongly subanalytic sets introduced in [11] as well as for those subanalytic sets considered in [6]. Let X ⊂ R1 be subanalytic. In [8], we showed that there is a decomposition of X as a union of a finite number of special sets U ⊂ R1 (see below). In this note, in Theorem 1.6, we obtain a version of this result which is uniform in parameters, thereby answering a question brought to our attention by Angus Macintyre. It follows immediately from Theorem 1.6 that the theory of K in the language (see [5] and [6]) is C-minimal in the sense of [3] and [9]. The analogous uniformity result in the p-adic case was recently proved in [12]. Definition 1.1. (i) A disc in R1 is a set of one of the two following forms: A special set in R1 is a disc minus a finite union of discs. (ii) R-domains u ⊂ Rm, and their associated rings of analytic functions, , are defined inductively as follows. Rm is an R-domain and , the ring of strictly convergent power series in X1,…, Xm over K. If u is an R-domain with associated ring , (where K 〈X, Y〉 〚ρ〛S is a ring of separated power series, see [5, §2] and [1, §1]) and f, have no common zero on u and ◸ ϵ {<, ≤}, then is an R-domain and where J is the ideal generated by I and f − gZ (Z is a new variable) if ◸ is ≤, and where J is the ideal generated by I and f − gτ (τ a new variable) if ◸ is <. (See [8, Definition 2.2].) R-domains generalize the rational domains of [2, §7.2.3]. It is true, but not easy to prove, that only depends on u as a point set, and is independent of the particular representation of u.