通用半单重构与障碍理论
Universal Semisimple Reconstruction and Obstruction Theory
- Dundalk Institute of Technology（邓多克理工学院）
机构由 AI 辅助整理，请以论文原文为准。
AI总结：
针对有限长度阿贝尔范畴建立半单障碍理论，构造典范障碍子对象与半单化反射子，其迭代可还原经典根基滤过，揭示经典根基理论是范畴框架的模论实例。
AI中文摘要：
我们针对有限长度阿贝尔范畴建立了半单障碍理论。对每个对象\boldsymbol{X}，我们关联一个典范障碍子对象\boldsymbol{\text{Ob}}(X)，它由所有到半单对象的态射无法触及的部分构成。商对象\boldsymbol{X/\text{Ob}(X)}是半单的，且在这类态射中具有泛性质，由此得到典范半单化反射子，并通过障碍消失来刻画半单性。迭代该构造可得到一个障碍滤过，在有限长度模范畴中，该滤过可还原经典根基滤过与Loewy理论。障碍还依赖于心：在Happel–Reiten–Smalø倾斜下，扩张数据可作为新的障碍层出现，进而产生障碍阴影与可见挠率。因此，经典根基理论是半单重构、心的持续存在与变化的范畴框架中固定心的模论实例。
英文摘要：
We develop a semisimple obstruction theory for finite-length abelian categories. To each object \(X\) we associate a canonical obstruction subobject \(\mathcal O(X)\), consisting of the part invisible to all morphisms into semisimple objects. The quotient \(X/\mathcal O(X)\) is semisimple and universal among such morphisms, yielding a canonical semisimplification reflector and characterizing semisimplicity by the vanishing of obstruction. Iterating the construction gives an obstruction filtration which, in finite-length module categories, recovers the classical radical filtration and Loewy theory. Obstruction is also heart-dependent: under Happel--Reiten--Smalø tilting, extension data can appear as new obstruction layers, leading to obstruction shadows and visible torsion. Thus classical radical theory arises as the fixed-heart module-theoretic instance of a categorical framework for semisimple reconstruction, persistence, and variation of heart.
