Herr complex of (φ,τ)-modules

Abstract

Let p be an odd prime number and K a mixed characteristic complete discrete valuation field with perfect residue field of characteristic p. We construct a three-term complex, defined in terms of the $(\phi ,\tau )$-module of a p-adic representation and prove its homology is isomorphic to the Galois cohomology of the representation. We further show that our complex is quasi-isomorphic to the four-term complex constructed by Tavares Ribeiro, providing an alternative proof of our result. As an application, we describe Galois cohomology of the Tate module associated to a p-divisible group in terms of corresponding Breuil-Kisin modules.