Compatibility of Kisin modules for different uniformizers

TL;DR

This paper demonstrates that Kisin modules associated with a semi-stable Galois representation are essentially independent of the choice of uniformizer, and shows how Wach modules can be recovered from (,  G)-modules in certain cases.

Contribution

It proves the isomorphism of Kisin modules for different uniformizers and establishes independence of certain lattices from uniformizer choice, also recovering Wach modules from (,  G)-modules.

Findings

Kisin modules are isomorphic for different uniformizers after tensoring.

Lattices in the filtered (, N)-module are uniformizer-independent.

Wach modules can be recovered from (,  G)-modules in unramified crystalline cases.

Abstract

Let p be a prime and T a lattice inside a semi-stable representation V. We prove that Kisin modules associated to T by selecting different uniformizers are isomorphic after tensoring a subring in W(R). As consequences, we show that several lattices inside the filtered (phi, N)-module of V constructed from Kisin modules are independent on the choice of uniformizers. Finally we use a similar strategy to show that the Wach module can be recovered from the (\varphi, \hat G)-module associated to T when V is crystalline and the base field is unramified.