Terminal Ambiguity Geometry for Finite-Dimensional Algebras
Abstract
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.
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BibTeX
@misc{gildea2026terminal,
title = {{Terminal Ambiguity Geometry for Finite-Dimensional Algebras}},
author = {Joe Gildea},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2609.24924},
eprint = {2609.24924},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.24924},
}
RIS
TY - UNPB
TI - Terminal Ambiguity Geometry for Finite-Dimensional Algebras
AU - Joe Gildea
PY - 2026
DA - 2026/09/21
PB - arXiv
DO - 10.48550/arxiv.2609.24924
UR - https://arxiv.org/abs/2609.24924
AB - 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.
ER -