We prove that for any computable successor ordinal of the form [Formula: see text] [Formula: see text] limit and [Formula: see text] there exists computable torsion-free abelian group [Formula: see text]TFAG[Formula: see text] that is relatively [Formula: see text] -categorical and not [Formula: see text] -categorical. Equivalently, for any such [Formula: see text] there exists a computable TFAG whose initial segments are uniformly described by [Formula: see text] infinitary computable formulae up to automorphism (i.e. it has a c.e. uniformly [Formula: see text]-Scott family), and there is no syntactically simpler (c.e.) family of formulae that would capture these orbits. As far as we know, the problem of finding such optimal examples of (relatively) [Formula: see text]-categorical TFAGs for arbitrarily large [Formula: see text] was first raised by Goncharov at least 10 years ago, but it has resisted solution (see e.g. Problem 7.1 in survey [Computable abelian groups, Bull. Symbolic Logic 20(3) (2014) 315–356]). As a byproduct of the proof, we introduce an effective functor that transforms a [Formula: see text]-computable worthy labeled tree (to be defined) into a computable TFAG. We expect that this technical result will find further applications not necessarily related to categoricity questions.

中文翻译：

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具有最优 Scott 族的无扭阿贝尔群

我们证明对于形式 [公式：见文本] [公式：见文本] 极限和 [公式：见文本] 的任何可计算后继序数，存在可计算无扭阿贝尔群 [公式：见文本]TFAG[公式：见text] 是相对的 [Formula: see text] -categorical 而不是 [Formula: see text] -categorical。等效地，对于任何这样的[公式：见文本]，都存在一个可计算的 TFAG，其初始段由 [公式：见文本] 直到自同构的无限可计算公式统一描述（即，它统一具有 ce [公式：见文本]-Scott族），并且没有语法上更简单（ce）的公式族可以捕获这些轨道。据我们所知，寻找（相对）[公式：见正文]的最佳示例的问题 - 任意大的 [公式：见文本] 至少在 10 年前由 Goncharov 首次提出，但它拒绝解决（参见调查中的问题 7.1 [可计算阿贝尔群，Bull. Symbolic Logic 20(3) (2014) 315–356]）。作为证明的副产品，我们引入了一个有效的函子，它将 [公式：见文本]-可计算的有价值的标记树（待定义）转换为可计算的 TFAG。我们期望这一技术结果将找到与分类问题不一定相关的进一步应用。