Henselian值域中的传递原理

Pierre Touchard
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We show, for instance, that the Hahn field \n$\\mathbb {F}_p^{\\text {alg}}((\\mathbb {Z}[1/p]))$\n is inp-minimal (of burden 1), and that the ring of Witt vectors \n$W(\\mathbb {F}_p^{\\text {alg}})$\n over \n$\\mathbb {F}_p^{\\text {alg}}$\n is not strong (of burden \n$\\omega $\n ). This result extends previous work by Chernikov and Simon and realizes an important step toward the classification of Henselian valued fields of finite burden. Second, we show a transfer principle for the property that all types realized in a given elementary extension are definable. It can be written as follows: a valued field as above is stably embedded in an elementary extension if and only if its value group is stably embedded in the corresponding extension of value groups, its residue field is stably embedded in the corresponding extension of residue fields, and the extension of valued fields satisfies a certain algebraic condition. 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引用次数: 1

摘要

摘要本文研究了具有等特征$0$的Henselian值场、代数闭值场、代数极大Kaplansky值场和具有完美残数场的未分枝混合特征Henselian值场的转移原理。首先,根据其值群和剩余域的负担计算该值域的负担。该负担是与模型理论复杂性和与$\text {NTP}_2$理论相关的维度概念相关的基数。例如,我们证明了Hahn域$\mathbb {F}_p^{\text {alg}}((\mathbb {Z}[1/p]))$是最小值(负荷1),并且Witt向量$W(\mathbb {F}_p^{\text {alg}})$超过$\mathbb {F}_p^{\text {alg}}$的环不是强(负荷$\omega $)。这一结果扩展了Chernikov和Simon先前的工作,并向有限负荷的Henselian值域的分类迈出了重要的一步。其次,我们展示了在给定的初等扩展中实现的所有类型都是可定义的属性的传递原理。可以写成:上式的值域稳定嵌入初等扩展,当且仅当其值群稳定嵌入值群的相应扩展,其剩余域稳定嵌入剩余域的相应扩展,且值域的扩展满足一定的代数条件。例如,我们证明幂级数字段$\mathbb {R}((t))$上的所有类型都是可定义的。类似地,$W(\mathbb {F}_p^{\text {alg}})$的商域上的所有类型都是可定义的。这扩展了库比德斯和德龙以及库比德斯和叶之前的工作。这些不同的结果使用了一种共同的方法,这种方法是最近发展起来的。它包括首先建立一个称为先导项结构的中间结构的约简,或者$\operatorname {\mathrm {RV}}$ -sort,然后约简到值组和剩余字段。这导致我们在阿贝尔群的纯短精确序列的背景下发展类似的约简原理。摘要由Pierre Touchard准备。电子邮件:pierre.pa.touchard@gmail.com URL: https://miami.uni-muenster.de/Record/a612cf73-0a2f-42c4-b1e4-7d28934138a9
本文章由计算机程序翻译,如有差异,请以英文原文为准。
Transfer Principles in Henselian Valued Fields
Abstract In this thesis, we study transfer principles in the context of certain Henselian valued fields, namely Henselian valued fields of equicharacteristic $0$ , algebraically closed valued fields, algebraically maximal Kaplansky valued fields, and unramified mixed characteristic Henselian valued fields with perfect residue field. First, we compute the burden of such a valued field in terms of the burden of its value group and its residue field. The burden is a cardinal related to the model theoretic complexity and a notion of dimension associated to $\text {NTP}_2$ theories. We show, for instance, that the Hahn field $\mathbb {F}_p^{\text {alg}}((\mathbb {Z}[1/p]))$ is inp-minimal (of burden 1), and that the ring of Witt vectors $W(\mathbb {F}_p^{\text {alg}})$ over $\mathbb {F}_p^{\text {alg}}$ is not strong (of burden $\omega $ ). This result extends previous work by Chernikov and Simon and realizes an important step toward the classification of Henselian valued fields of finite burden. Second, we show a transfer principle for the property that all types realized in a given elementary extension are definable. It can be written as follows: a valued field as above is stably embedded in an elementary extension if and only if its value group is stably embedded in the corresponding extension of value groups, its residue field is stably embedded in the corresponding extension of residue fields, and the extension of valued fields satisfies a certain algebraic condition. We show, for instance, that all types over the power series field $\mathbb {R}((t))$ are definable. Similarly, all types over the quotient field of $W(\mathbb {F}_p^{\text {alg}})$ are definable. This extends previous work of Cubides and Delon and of Cubides and Ye. These distinct results use a common approach, which has been developed recently. It consists of establishing first a reduction to an intermediate structure called the leading term structure, or $\operatorname {\mathrm {RV}}$ -sort, and then of reducing to the value group and residue field. This leads us to develop similar reduction principles in the context of pure short exact sequences of abelian groups. Abstract prepared by Pierre Touchard. E-mail: pierre.pa.touchard@gmail.com URL: https://miami.uni-muenster.de/Record/a612cf73-0a2f-42c4-b1e4-7d28934138a9
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