ETD

• cotangent complex
• derived algebra
• derived category
• derived de Rham
• divided powers
• homological algebra
• p-adic Hodge theory
• period rings

We construct the de Rham, crystalline and semistable period rings of Fontaine using derived de Rham theory, following the work of Beilinson and Bhatt.
We use model categories and simplicial methods to define the cotangent complex and the derived de Rham algebra for morphisms of both ordinary and pre-logarithmic algebras, and we study their properties.
The theory of deformations of algebras of Illusie also plays a key role in this work, as the connection between deformations and the cotangent complex will allow us to explain the universal property of period rings as $p$-adic thickenings of the rings of integers of a completed algebraic closure of a $p$-adic field.