The canonical dimension of modules for Iwasawa algebras

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 $G{L}_n(F)$, with central character, has finite length if its dual has canonical dimension two.

37 pages. Primary results extended to semisimple Lie algebras. Section 7 rewritten to give an improved statement. Section 8 expanded to improve an abstract bound on Krull dimension. Section 9 shortened by a new proof that holonomic representations are of finite length. Minor corrections and edits throughout. Acknowledgements updated. Manuscript accepted in Selecta Mathematica

References in corpus (1)

• Coherence of Augmented Iwasawa Algebras