Generalized Robba rings

TL;DR

This paper proves that projective coadmissible modules over the distribution algebra of a compact p-adic Lie group are finitely generated, introduces a generalized Robba ring for uniform pro-p groups, and explores its topological properties.

Contribution

It constructs a generalized Robba ring containing the distribution algebra and develops microlocalization theory for quasi-abelian normed algebras, advancing the understanding of coadmissible modules.

Findings

Projective coadmissible modules are finitely generated.

The generalized Robba ring contains the distribution algebra as a subring.

Topological properties of the generalized Robba ring are established.

Abstract

We prove that any projective coadmissible module over the locally analytic distribution algebra of a compact $p$-adic Lie group is finitely generated. In particular, the category of coadmissible modules does not have enough projectives. In the Appendix a "generalized Robba ring" for uniform pro-$p$ groups is constructed which naturally contains the locally analytic distribution algebra as a subring. The construction uses the theory of generalized microlocalization of quasi-abelian normed algebras that is also developed there. We equip this generalized Robba ring with a self-dual locally convex topology extending the topology on the distribution algebra. This is used to show some results on coadmissible modules.