Cohomology of ($\phi$, $\Gamma$)-modules and duality

Cohomology of ($\phi$, $\Gamma$)-modules was studied by Herr, Liu, and Kedlaya-Pottharst-Xiao. Kedlaya-Pottharst-Xiao proved finiteness, duality, and Euler-characteristic formula for cohomology of families of ($\phi$, $\Gamma$)-modules.

In this talk, we will present an alternative proof of finiteness and duality by using analytic geometry introduced by Clausen-Scholze and 6-functor formalism refined by Heyer-Mann. One advantage of this proof is that it can handle families over Banach Qp-algebras that are not topologically of finite type over Qp. If time permits, we will also discuss potential future applications to the representability of the analytic Emerton-Gee stack.