On definable groups and D-groups in certain fields with a generic derivation

Canadian Journal of Mathematics Pub Date : 2024-01-15 DOI:10.4153/s0008414x24000063

Ya’acov Peterzil, Anand Pillay, Françoise Point

引用次数: 0

Abstract

We continue our study from Peterzil et al. (2022, Preprint, arXiv:2208.08293) of finite-dimensional definable groups in models of the theory Abstract Image $T_{\partial }$, the model companion of an o-minimal ${\mathcal {L}}$-theory T expanded by a generic derivation $\partial $ as in Fornasiero and Kaplan (2021, Journal of Mathematical Logic 21, 2150007).

We generalize Buium’s notion of an algebraic D-group to Abstract Image ${\mathcal {L}}$-definable D-groups, namely $(G,s)$, where G is an ${\mathcal {L}}$-definable group in a model of T, and $s:G\to \tau (G)$ is an ${\mathcal {L}}$-definable group section. Our main theorem says that every definable group of finite dimension in a model of $T_\partial $ is definably isomorphic to a group of the form Abstract Image $$ \begin{align*}(G,s)^\partial=\{g\in G:s(g)=\nabla g\},\end{align*} $$

for some Abstract Image ${\mathcal {L}}$-definable D-group $(G,s)$ (where $\nabla (g)=(g,\partial g)$).

We obtain analogous results when T is either the theory of p-adically closed fields or the theory of pseudo-finite fields of characteristic Abstract Image $0$.

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关于某些域中的可定义群和 D 群的通用推导

我们继续研究彼得齐尔等人（2022，预印本，arXiv:2208.08293）理论模型中的有限维可定义群$T_{\partial }$，即一个o-最小${\mathcal {L}}$理论T的模型同伴，该理论由通用派生$\partial $展开，如福纳西耶罗和卡普兰（2021，《数理逻辑杂志》21，2150007）。我们将布伊姆的代数 D 群概念推广到 ${\mathcal {L}}$ 可定义 D 群，即 $(G,s)$，其中 G 是 T 模型中的 ${mathcal {L}}$ 可定义群，而 $s:G\to \tau (G)$ 是 ${\mathcal {L}}$ 可定义群部分。我们的主要定理指出，在 $T_\partial $ 的模型中，每一个有限维的可定义群都与形式为 $$ \begin{align*}(G,s)^\partial=\{g\in G:s(g)=\nabla g\},\end{align*} 的群同构。对于某个 ${mathcal {L}}$ 定义的 D 群 $(G,s)$（其中 $\nabla (g)=(g,\partial g)$），我们会得到类似的结果。当 T 是 p-adically closed fields 理论或特征 $0$ 的伪无限域理论时，我们会得到类似的结果。

本文章由计算机程序翻译，如有差异，请以英文原文为准。

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Canadian Journal of Mathematics

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