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

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