Appears in collection : Ahmed Abbes - The p-adic Simpson correspondence: Functoriality by proper direct image and Hodge-Tate local systems

Faltings initiated in 2005 a p-adic analogue of the (complex) Simpson correspondence whose construction has been taken up by various authors and whose properties have been developed according to several approaches. I will present in these lectures the approach I developed with Michel Gros, inspired by the Cartier transform of Ogus and Vologodsky, which is an analogue in characteristic p of Simpson correspondence. The p-adic Simpson correspondence can be considered as a categorification of the Hodge-Tate decomposition. I will present its construction for small generalized representations using a suitable period ring, establish its functoriality by proper direct image and discuss the link with Hodge-Tate local systems.