发表机构
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 中文总结
本文针对有限维代数的根截断导致模信息丢失的问题，引入终端歧义几何，通过格拉斯曼轨道空间分类终端完备化，并给出刚性判据、维数及有限域计数，同时建立重构与对称性结果。
AI 中文摘要
根截断会遗忘关于模的信息。我们研究这种丢失所产生的歧义：在所有适当的根截断被固定之后，哪些非同构的模仍然无法区分，由此产生的族如何组织，以及可以从它们中恢复哪些信息。我们引入终端歧义几何，将这种残余的不确定性转化为几何对象。剩余的歧义由终端根层控制，我们通过格拉斯曼轨道空间对终端完备化进行分类，其中射影歧义作为秩一情形出现。该分类给出了刚性和非唯一性的判据，并且在分裂情形下，确定了完备化空间的维数，并给出了非同构终端完备化的精确有限域计数。我们还建立了秩一射影几何的重构和对称性结果。在适当的秩假设下，歧义几何恢复了相应的剩余除代数数据、终端种类和终端根层，而其自同构由经典射影半线性群描述。因此，本文研究了根截断的两个方面：其遗忘的信息所产生的歧义，以及从所得几何中恢复该信息的程度。
英文摘要
Radical truncation forgets information about a module. We study the ambiguity created by this loss: which nonisomorphic modules remain indistinguishable after all proper radical truncations have been fixed, how the resulting families are organized, and what information can be recovered from them. We introduce terminal ambiguity geometry to make this residual indeterminacy into a geometric object. The remaining ambiguity is governed by the terminal radical layer, and we classify terminal completions by Grassmannian orbit spaces, with projective ambiguity appearing as the rank-one case. This classification gives criteria for rigidity and nonuniqueness and, in the split case, determines the dimensions of the completion spaces and gives exact finite-field counts of nonisomorphic terminal completions. We also establish reconstruction and symmetry results for the rank-one projective geometry. Under suitable rank hypotheses, the ambiguity geometry recovers the corresponding residue division-algebra data, terminal species, and terminal radical layer, while its automorphisms are described by classical projective semilinear groups. Thus the paper studies both sides of radical truncation: the ambiguity produced by the information it forgets and the extent to which that information can be recovered from the resulting geometry.
Comments20 pages
