Year: 2026
Type: article
Abstract: Let $${\mathcal {F}} \, = \, (\dots {\mathop {\rightarrow }\limits ^{\partial _{n+1}}} {\mathcal {F}}_n {\mathop {\rightarrow }\limits ^{\partial _n}} {\mathcal {F}}_{n-1}{\mathop {\rightarrow }\limits ^{\partial _{n-1}}} \dots \dots {\mathop {\rightarrow }\limits ^{\partial _1}} {\mathcal {F}}_0 \rightarrow {\mathfrak {R}} \rightarrow 0)$$ F = ( ⋯ → ∂ n + 1 F n → ∂ n F n - 1 → ∂ n - 1 ⋯ ⋯ → ∂ 1 F 0 → R → 0 ) be a free resolution over the group ring $${\mathfrak {R}}[\Phi ]$$ R [ Φ ] where $${\mathfrak {R}}$$ R is commutative and $$\Phi $$ Φ is finite. The $$n^{th}$$ n th syzygy $$\Omega _n^{{\mathfrak {R}}[\Phi ]}$$ Ω n R [ Φ ] is the stable class of $$\textrm{Im}(\partial _n)$$ Im ( ∂ n ) and has a tree structure with roots which do not extend infinitely downwards. We show that $$\Omega _3^{{\mathfrak {R}}[Q_{8p}]}$$ Ω 3 R [ Q 8 p ] has infinitely many isomorphically distinct modules at the minimal level when $$\,{\mathfrak {R}} = {\mathbb {Z}}[C_\infty ]$$ R = Z [ C ∞ ] is the integral group ring of the infinite cyclic
Author: F. E. A. Johnson
Institution: University College London
Language: English
Subfield: Algebra and Number Theory
Field: Mathematics
Domain: Physical Sciences
Open Access status: hybrid
Estimated APC paid: $3,190
OpenAlex ID: W7128803631