The short exact sequence in definable Galois cohomology

David Meretzky
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Abstract

In Remarks on Galois Cohomology and Definability [2], Pillay introduced definable Galois cohomology, a model-theoretic generalization of Galois cohomology. Let $M$ be an atomic and strongly $\omega$-homogeneous structure over a set of parameters $A$. Let $B$ be a normal extension of $A$ in $M$. We show that a short exact sequence of automorphism groups $1 \to \text{Aut}(M/B) \to \text{Aut}(M/A) \to \text{Aut}(B/A) \to 1$ induces a short exact sequence in definable Galois cohomology. Our result complements the long exact sequence in definable Galois cohomology developed in More on Galois cohomology, definability and differential algebraic groups [3].
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可定义伽罗瓦同调中的短精确序列
在《关于伽罗瓦同调与可定义性的评论》[2]中,皮莱介绍了可定义伽罗瓦同调,它是伽罗瓦同调的一种模型理论概括。让 $M$ 是在一组参数 $A$ 上的原子和强 $\omega$ 同调结构。让 $B$ 是 $A$ 在 $M$ 中的正常扩展。我们可以看到,自变群的短精确序列 $1 (到)\text{Aut}(M/B)(到)\text{Aut}(M/A)(到)\text{Aut}(B/A)(到)1$ 在可定义的伽罗瓦同调中诱导了一个短精确序列。我们的结果是对《更多关于伽罗瓦同调、可定义性和微分代数群》[3] 中提出的可定义伽罗瓦同调中的长精确序列的补充。
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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