Cohomology of $(\varphi,\Gamma)$-modules over pseudorigid spaces

We study the cohomology of families of $(\varphi,\Gamma)$-modules with coefficients in pseudoaffinoid algebras. We prove that they have finite cohomology, and we deduce an Euler characteristic formula and Tate local duality. We classify rank-$1$ $(\varphi, \Gamma)$-modules and deduce that triangulations of pseudorigid families of $(\varphi,\Gamma)$-modules can be interpolated, extending a result of [KPX14]. We then apply this to study extended eigenvarieties at the boundary of weight space, proving in particular that the eigencurve is proper at the boundary and that Galois representations attached to certain characteristic $p$ points are trianguline.