Compatibility of Kisin modules for different uniformizers

Abstract

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 {(\varphi,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.