P-adic Hodge Theory And P-adic Simpson Correspondence For Rigid Analytic Varieties

This paper serves as a learning note about the recent progress of p-adic Hodge theory and p-adic Simpson correspondence for rigid analytic varieties.We begin by reviewing Huber’s adic space,which serves as the fundamental geometric object in this paper.Next we introduce the recent breakthrough on p-adic Hodge theory of rigid analytic varieties achieved by Scholze.Specifically,along the idea of Scholze,we prove some comparison theorems between different cohomology theories of rigid analytic varieties,the key notions in the proof are the proetale site and the period sheaves on this site.Finally,based on Scholze’s theory,after reviewing the classical Simpson correspondence over C and Faltings’ p-adic analogue,we introduce Ruochuan Liu and Xinwen Zhu’s work on p-adic Simpson correspondence and give a sketch of their proof.