关于全超越环境中的可定义伽罗瓦理论与可定义伽罗瓦上同调
On definable Galois theory and definable Galois cohomology in the totally transcendental setting
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中文总结 AI 辅助
本文在全超越一阶理论中建立可定义伽罗瓦理论与上同调的联系，给出微分伽罗瓦上同调的余极限公式，并引入微分域和参数集的有界性新概念。
中文摘要 AI 辅助
本文给出了在全超越一阶理论环境中，将文献[28]的可定义伽罗瓦理论与文献[18]的可定义伽罗瓦上同调联系起来的结果。首先，我们证明在一些常见假设下，与固定内部性数据相关联的外在可定义伽罗瓦群的可定义伽罗瓦上同调，对素数模型的一个副本内部的可定义伽罗瓦扩张的数量进行了分类。这推广了一个著名的结果，即Picard-Vessiot微分伽罗瓦理论中的相应结果，该结果最初通过tannakian方法证明[3]。然后，我们汇集了微分伽罗瓦上同调的一些平凡性结果[21]以及关于迭代Picard-Vessiot扩张的近期结果[10][15]，以给出微分伽罗瓦上同调的余极限公式。具体而言，对于特征为0的常微分域$K$和$K$上的线性微分代数群$G$，微分伽罗瓦上同调$\text{H}^1_{\delta}(K,G)$由$K$的有限型迭代Picard-Vessiot扩张的正规闭包族上的余极限给出。接着，我们提出了微分域有界性的一个新概念。最后，我们证明了参数集$A$的最小闭包，即$A$上素数模型的一个副本到其自身的所有初等嵌入的交集，包含了$A$上满足一些强但常见条件的可定义群的所有可定义伽罗瓦上同调信息。最后，我们给出可定义伽罗瓦上同调的一些余极限公式，并提出了参数集有界性的模型论定义。
英文摘要
This paper gives results which relate the definable Galois theory of [28] to the definable Galois cohomology of [18] in the setting of a totally transcendental first order theory. Firstly, we show that under some common assumptions the definable Galois cohomology of the extrinsic definable Galois group associated to fixed internality data classifies the number of definable Galois extensions inside a copy of the prime model. This generalizes a well-known result for Picard-Vessiot differential Galois theory shown originally via tannakian methods [3]. We then collate some triviality results for differential Galois cohomology [21] and recent results on iterated Picard-Vessiot extensions [10] [15] to give colimit formulas for differential Galois cohomology. Precisely, for an ordinary differential field $K$ of characteristic $0$ and a linear differential algebraic group $G$ over $K$, the differential Galois cohomology $\text{H}^1_δ(K,G)$, is given by a colimit over the family of normal closures of iterated Picard-Vessiot extensions of $K$ of finite type. We then propose a new notion of boundedness for a differential field. We finally show that the minimal closure of a set of parameters $A$, the intersection of all elementary embeddings of a copy of the prime model over $A$ into itself, contains all of the definable Galois cohomological information for definable groups over $A$ satisfying also some strong but common conditions. Lastly, we give some colimit formulas for definable Galois cohomology and propose a model-theoretic definition of a bounded set of parameters.
发表机构
- Universidad de Los Andes（洛斯安第斯大学）
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