On the continuity of the finite Bloch–Kato cohomology

Abstract

Let $K_0$ be an unramified, complete discrete valuation field of mixed characteristics $(0,p)$ with perfect residue field. We consider two finite, free ${\mathbb{Z}}_p$-representations of $G_{K_0}$, $T_1$ and $T_2$, such that $T_i{\otimes }_{{\mathbb{Z}}_p}{\mathbb{Q}}_p$, for $i=1,2$, are crystalline representations with Hodge-Tate weights between $0$ and $r\le p-2.$ Let $K$ be a totally ramified extension of degree $e$ of $K_0$. Supposing that $p\ge 3$ and $e(r-1)\le p-1$, we prove that for every integer $n\ge 1$ and $i=1,2$, the inclusion $H_f^1(K,T_i)/p^n{H}_f^1(K,T_i)\hookrightarrow H^1(K,T_i/p^n{T}_i)$ of the finite Bloch-Kato cohomology into the Galois cohomology is functorial with respect to morphisms as $\mathbb{Z}/p^n\mathbb{Z}[G_{K_0}]$-modules from $T_1/p^n{T}_1$ to $T_2/p^n{T}_2$. In the appendix we give a related result for $p=2$.