The direct summand theorem

In this thesis, our aim is to present a recent result proved by Yves André: Hochster's direct summand conjecture in commutative algebra; we will focus on the approach given by Bhargav Bhatt in his article, in which he streamlines André's original proof. Hochster's original conjecture asks if, given a finite extension of noetherian rings $i:A_0\rightarrow B_0$ with $A_0$ regular, the inclusion splits as a map of $A_0$-modules. Hochster himself reduced the problem to the case of $A_0$ being local and complete, and solved the conjecture in the case of unmixed characteristics. The first chapter will focus on these simplifications, and will contain an easy proof of the unmixed case in characteristics $0$. André and Bhatt proved the remaining case of characteristics $(0,p)$, which will be the focus of the rest of the this thesis. It turns out that only working with $A_0$-modules is too stringent: Faltings' almost mathematics allows us to consider more flexible categories of almost-modules. This theory will be explored in chapter 2. We will introduce Scholze's perfectoid theory in chapter 3. It will allow us to talk about Faltings' almost purity theorem, and construct a suitable extension $A$ for its application. Finally, chapter 4 will contain a brief description of pro-modules and almost-pro-modules, while chapter 5 will tackle the main theorem, after presenting some additional useful lemmas