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

@inproceedings{Bellovin2021CohomologyO, title={Cohomology of \$(\varphi,\Gamma)\$-modules over pseudorigid spaces}, author={Rebecca Bellovin}, year={2021} }

. We study the cohomology of families of ( ϕ, Γ)-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 ( ϕ, Γ)-modules and deduce that triangulations of pseudorigid families of ( ϕ, Γ)-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…

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