Multivariable Lubin-Tate (\phi,\Gamma)-modules and filtered \phi-modules

TL;DR

This paper introduces multivariable Lubin-Tate (,\u0393)-modules, constructs associated rings and sheaves, and proves that all crystalline (,)-modules over these rings can be obtained via a specific modification process.

Contribution

It extends the theory of (,)-modules to multivariable settings and characterizes crystalline modules through a novel construction.

Findings

Defined new rings of power series in multiple variables.

Constructed (,)-modules as sections of reflexive sheaves.

Proved all crystalline (,)-modules arise from the proposed construction.

Abstract

We define some rings of power series in several variables, that are attached to a Lubin-Tate formal module. We then give some examples of (\phi,\Gamma)-modules over those rings. They are the global sections of some reflexive sheaves on the p-adic open unit polydisk, that are constructed from a filtered \phi-module using a modification process. We prove that we obtain every crystalline (\phi,\Gamma)-module over those rings in this way.