Cartier smoothness in prismatic cohomology

TL;DR

This paper introduces the concept of p-Cartier smooth algebras, generalizing smooth algebras, and explores their properties and relationships within prismatic cohomology, including a comparison theorem between syntomic and étale cohomologies.

Contribution

It defines p-Cartier smoothness, provides characterizations via prismatic cohomology, and establishes a comparison theorem between syntomic and étale cohomologies.

Findings

p-Cartier smooth algebras include valuation rings over perfectoid bases

Characterizations of p-Cartier smoothness in terms of prismatic cohomology

A comparison theorem between syntomic and étale cohomologies

Abstract

We introduce the notion of a $p$-Cartier smooth algebra. It generalises that of a smooth algebra and includes valuation rings over a perfectoid base. We give several characterisations of $p$-Cartier smoothness in terms of prismatic cohomology, and deduce a comparison theorem between syntomic and \'etale cohomologies under this hypothesis.