On Emerton's $p$-adic Banach spaces

TL;DR

This paper introduces new methods for analyzing Emerton's $p$-adic Banach spaces, relating them to sheaf cohomology, generalizing spectral sequences, and deriving vanishing theorems.

Contribution

It provides a novel approach to study $p$-adic Banach spaces by connecting them with sheaf cohomology and extends Emerton's spectral sequence.

Findings

Established relations between $p$-adic Banach spaces and sheaf cohomology

Generalized Emerton's spectral sequence

Derived vanishing theorems in specific cases

Abstract

The purpose of the current paper is to introduce some new methods for studying the $p$-adic Banach spaces introduced by Emerton \cite{emerton}. We first relate these spaces to more familiar sheaf cohomology groups. As an application, we obtain a more general version of Emerton's spectral sequence. We also calculate the spaces in some easy cases. As a consequence, we obtain a number of vanishing theorems.