Cohomology of $(\varphi, \tau)$-modules

Abstract

We construct cohomology theories for $(\varphi, \tau)$-modules, and study their relation with cohomology of $(\varphi, \Gamma)$-modules, as well as Galois cohomology. Our method is axiomatic, and can treat the \'etale case, the overconvergent case, and the rigid-overconvergent case simultaneously. We use recent advances in locally analytic cohomology as a key ingredient.