论文标题
无扭转的阿贝尔群体扩展的分类问题，我
The classification problem for extensions of torsion-free abelian groups, I
论文作者
论文摘要
令$ c，$是可计数的阿贝尔团体。在本文中，我们确定了将扩展名$ c $按$ a $进行分类的复杂性，在$ c $不含扭转的情况下，$ a $是$ p $ group，一个具有有限的主组件的扭转组，或一个免费的$ r $ - module，用于某些子$ r \ subseteq \ subseteq \ subseteq \ mathbb {q} $。 Precisely, for such $C$ and $A$ we describe in terms of $C$ and $A$ the potential complexity class in the sense of Borel complexity theory of the equivalence relation $\mathcal{R}_{\mathbf{Ext}\left( C,A\right) }$ of isomorphism of extensions of $C$ by $A$.这补充了同一位作者的先前结果，该结果在$ c $是扭转时解决了案例，而$ a $是任意的。我们在最近与Bergfalk和Panagiotopoulos合作推出的Borel可定义同源代数的框架内建立了主要结果。由于我们的主要结果，我们将获得，如果$ c $是无扭转的，而$ a $是一个免费的$ r $ - 模块，则是具有有界组件的扭转组，则在$ a $ a $ a splits的延伸时，只有在所有有限股票子群体上分配给所有有限股票的$ c $ $ c $。这是一种纯粹的代数陈述，该陈述是通过可从鲍尔可定义同源代数的方法获得的。
Let $C,A$ be countable abelian groups. In this paper we determine the complexity of classifying extensions $C$ by $A$, in the cases when $C$ is torsion-free and $A$ is a $p$-group, a torsion group with bounded primary components, or a free $R$-module for some subring $R\subseteq \mathbb{Q}$. Precisely, for such $C$ and $A$ we describe in terms of $C$ and $A$ the potential complexity class in the sense of Borel complexity theory of the equivalence relation $\mathcal{R}_{\mathbf{Ext}\left( C,A\right) }$ of isomorphism of extensions of $C$ by $A$. This complements a previous result by the same author, settling the case when $C$ is torsion and $A$ is arbitrary. We establish the main result within the framework of Borel-definable homological algebra, recently introduced in collaboration with Bergfalk and Panagiotopoulos. As a consequence of our main results, we will obtain that if $C$ is torsion-free and $A$ is either a free $R$-module or a torsion group with bounded components, then an extension of $C$ by $A$ splits if and only if it splits on all finite-rank subgroups of $C$. This is a purely algebraic statements obtained with methods from Borel-definable homological algebra.
