Iwasawa cohomology of analytic $(\varphi _L,{\varGamma }_L)$-modules

We show that the coadmissibility of the Iwasawa cohomology of an $L$-analytic Lubin–Tate $(\varphi_L,\Gamma_L)$-module $M$ is necessary and sufficient for the existence of a comparison isomorphism between the former and the analytic cohomology of its Lubin–Tate deformation, which, roughly speaking, is given by the base change of $M$ to the algebra of $L$-analytic distributions. We verify that coadmissibility is satisfied in the trianguline case and show that it can be “propagated” to a reasonably large class of modules, provided it can be proven in the étale case.