Compatibility of Kisin modules for different uniformizers

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®. 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.