G_a^{perf}-modules and de Rham Cohomology

TL;DR

This paper demonstrates that algebraic de Rham cohomology over smooth _p-algebras is formally e9tale, establishing a link to crystalline cohomology through new algebraic structures called pointed _a^{ ext{perf}}-modules.

Contribution

It introduces the notion of pointed _a^{ ext{perf}}-modules and quasi-ideals, providing a new framework to understand de Rham cohomology as a functorial deformation of crystalline cohomology.

Findings

Proves de Rham cohomology is formally e9tale.

Defines pointed _a^{ ext{perf}}-modules and quasi-ideals.

Redefines de Rham cohomology using new algebraic structures.

Abstract

We prove that algebraic de Rham cohomology as a functor defined on smooth $\mathbb{F}_p$-algebras is formally \'etale in a precise sense. This result shows that given de Rham cohomology, one automatically obtains the theory of crystalline cohomology as its unique functorial deformation. To prove this, we define and study the notion of a pointed $\mathbb{G}_a^{\mathrm{perf}}$-module and its refinement which we call a quasi-ideal in $\mathbb{G}_a^{\mathrm{perf}}$ -- following Drinfeld's terminology. Our main constructions show that there is a way to "unwind" any pointed $\mathbb{G}_a^{\text{perf}}$-module and define a notion of a cohomology theory for algebraic varieties. We use this machine to redefine de Rham cohomology theory and deduce its formal \'etalness and a few other properties.