Hodge completed derived de Rham algebra of a perfect ring

TL;DR

This paper computes the Hodge completed derived de Rham complex for perfect rings, extending the understanding of derived de Rham cohomology in algebraic geometry and number theory contexts.

Contribution

It provides an explicit computation of the Hodge completed derived de Rham algebra for perfect rings, filling a gap in the understanding of derived de Rham cohomology for these rings.

Findings

Explicit description of the Hodge completed derived de Rham algebra for perfect rings.

Connections to p-adic periods and zeta function invariants.

Extension of Morin's results to a broader class of rings.

Abstract

Derived de Rham cohomology has been recently used in several contexts, as in works of Beilinson and Bhatt on p-adic periods morphisms and Morin on numerical invariants for special values of zeta functions. Inspired by some results of Morin, we aimed to compute Hodge completed derived de Rham complex in the case of a rings map $\mathbb{Z}\longrightarrow k$, factoring through $\mathbb{F}_p$, with $k$ a perfect ring (i.e. the Frobenius map is an automorphism).