Galois representations over pseudorigid spaces

We study $p$-adic Hodge theory for families of Galois representations over pseudorigid spaces. Such spaces are non-archimedean analytic spaces which may be of mixed characteristic, and which arise naturally in the study of eigenvarieties at the boundary of weight space. We construct overconvergent $(\varphi,\Gamma)$-modules for Galois representations over pseudorigid spaces, and we show that such $(\varphi,\Gamma)$-modules have finite cohomology. As a consequence, we deduce that the cohomology groups yield coherent sheaves, and we give partial results extending triangulations defined away from closed subspaces.