Correspondance de Simpson p-adique II : fonctorialit\'e par image directe propre et syst\`emes locaux de Hodge-Tate

TL;DR

This paper advances the p-adic Simpson correspondence by establishing its functoriality via proper direct image and exploring its connection to Hodge-Tate local systems, expanding the theoretical framework initiated by Faltings.

Contribution

It introduces the functoriality of the p-adic Simpson correspondence through proper direct image and links it to Hodge-Tate local systems, expanding previous constructions.

Findings

Established functoriality of the p-adic Simpson correspondence

Connected the correspondence to Hodge-Tate local systems

Expanded the scope of the original construction

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 develop in this new monograph 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. Faltings a d\'egag\'e en 2005 un analogue p-adique de la correspondance de Simpson (complexe) dont la construction a \'et\'e reprise par diff\'erents auteurs, selon plusieurs approches. Poursuivant celle que nous avons initi\'ee pr\'ec\'edemment, nous d\'eveloppons dans la pr\'esente monographie de nouveaux aspects de la correspondance de Simpson p-adique, inspir\'es par notre construction de la suite spectrale de Hodge-Tate relative. Nous traitons tout d'abord du lien avec les syst\`emes locaux de Hodge-Tate. Nous \'etablissons ensuite la fonctorialit\'e de la correspondance de Simpson p-adique par image directe propre. Chemin faisant, nous \'elargissons la port\'ee de notre construction initiale.