Twisted calculus in several variables

TL;DR

This paper develops a formal framework for twisted differential operators in multiple variables, focusing on twisted coordinates in Huber rings, with implications for $p$-adic Hodge and prismatic cohomology.

Contribution

It introduces new concepts and establishes an equivalence between modules with twisted connections and twisted derivatives, advancing the theory of twisted differential operators.

Findings

Established an equivalence between modules with twisted connections and twisted derivatives

Analyzed convergence properties of twisted differential operators

Connected the framework to $p$-adic Hodge and prismatic cohomology

Abstract

In this paper, we introduce novel concepts and establish a formal framework for twisted differential operators in the context of several variables. The focus is on twisted coordinates within Huber rings, which facilitate the construction of diverse rings of twisted differential operators. We establish an equivalence between modules equipped with twisted connections and those endowed with an actions of twisted derivatives. Furthermore, we examine the convergence properties of twisted differential operators under specific conditions. This work aligns with the ongoing advancements, in $p$-adic Hodge cohomology and prismatic cohomology.