The canonical dimension of modules for Iwasawa algebras

2023/06/16 by James Timmins, Timmins, James
Mathematics · #16P60 #16P90 #22E50 #Advanced Algebra and Geometry #Algebraic Geometry and Number Theory #FOS: Mathematics #Homotopy and Cohomology in Algebraic Topology #Number Theory (math.NT) #Representation Theory (math.RT) #Rings and Algebras (math.RA)

paper · pdf · doi:10.48550/arxiv.2306.09696

openalex publication_date 2023/06/16 · openalex created_date 2025/10/10 · openalex updated_date 2026/08/02

Abstract

Let F be a non-trivial finite extension of the p-adic numbers, and G be a compact p-adic Lie group whose Lie algebra is isomorphic to a split semisimple F-Lie algebra. We prove that the mod p Iwasawa algebra of G has no modules of canonical dimension one. One consequence is a new upper bound on the Krull dimension of the Iwasawa algebra. We also prove a canonical dimension-theoretic criterion for a mod p smooth admissible representation to be of finite length. Combining our results shows that any smooth admissible representation of GL_n(F), with central character, has finite length if its dual has canonical dimension two.

Related

• Modules over Iwasawa algebras
• K₀-invariance of the completely faithful property of Iwasawa modules
• Faithful completely reducible representations of modular Lie algebras
• A canonical dimension estimate for non-split semisimple p-adic Lie groups
• K₀ and the dimension filtration for p-torsion Iwasawa modules