The p-adic Simpson Correspondence II: Functoriality by proper direct image and Hodge-Tate local systems -- an overview

TL;DR

This paper develops a new aspect of the p-adic Simpson correspondence, focusing on functoriality via proper direct images and its relation to Hodge-Tate local systems, expanding the theoretical framework of p-adic Hodge theory.

Contribution

It introduces a new functoriality property of the p-adic Simpson correspondence through proper direct images and explores its connection to Hodge-Tate local systems.

Findings

Established functoriality of the p-adic Simpson correspondence

Connected the correspondence to Hodge-Tate local systems

Expanded the scope of the p-adic Simpson framework

Abstract

Faltings initiated in 2005 a p-adic analogue of the (complex) Simpson correspondence whose construction has been taken up by various authors, according to several approaches. Following the one we initiated previously, we present an overview of a new monograph developing new features of the p-adic Simpson correspondence, inspired by our construction of the relative Hodge-Tate spectral sequence. First, we address the connection to Hodge-Tate local systems. Second, we establish the functoriality of the p-adic Simpson correspondence by proper direct image. Along the way, we expand the scope of our original construction.