Cohomology of arithmetic families of (𝜑,Γ)-modules

Abstract

We prove the finiteness and compatibility with base change of the ( φ , Γ ) (\varphi , \Gamma ) -cohomology and the Iwasawa cohomology of arithmetic families of ( φ , Γ ) (\varphi , \Gamma ) -modules. Using this finiteness theorem, we show that a family of Galois representations that is densely pointwise refined in the sense of Mazur is actually trianguline as a family over a large subspace. In the case of the Coleman-Mazur eigencurve, we determine the behavior at all points.