Formula omitted]-modules and de Rham cohomology

Abstract

We prove that algebraic de Rham cohomology as a functor defined on smooth Fp-algebras is formally étale 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 Gaperf-module and its refinement which we call a quasi-ideal in Gaperf – following Drinfeld's terminology. Our main constructions show that there is a way to “unwind” any pointed Gaperf-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 étalness and a few other properties.