The Fargues-Fontaine factorization theorem

The aim of this work is to present a complete and self-contained proof of the Fargues–Fontaine factorization theorem, a central result in $p$-adic Hodge theory. This theorem provides a canonical decomposition of elements in the period ring A_inf, based on the geometry of the associated Newton polygon. To lay the groundwork, we introduce the foundational notions of the tilting correspondence and Witt vectors.