Finiteness of analytic cohomology of Lubin-Tate $({\phi }_L,{\mathrm{\Gamma }}_L)$-modules

We prove finiteness and base change properties for analytic cohomology of families of $L$-analytic $({\phi }_L,{\mathrm{\Gamma }}_L)$-modules parametrised by affinoid algebras in the sense of Tate. For technical reasons we work over a field $K$ containing a period of the Lubin-Tate group, which allows us to describe analytic cohomology in terms of an explicit generalized Herr complex.