# On the continuity of the finite Bloch–Kato cohomology

### Adrian Iovita

Università di Padova, Italy### Adriano Marmora

Université de Strasbourg, France

## 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 $Z_{p}$-representations of $G_{K_{0}}$, $T_{1}$ and $T_{2}$, such that $T_{i}⊗_{Z_{p}}Q_{p}$, for $i=1,2$, are crystalline representations with Hodge-Tate weights between $0$ and $r≤p−2.$ Let $K$ be a totally ramified extension of degree $e$ of $K_{0}$. Supposing that $p≥3$ and $e(r−1)≤p−1$, we prove that for every integer $n≥1$ and $i=1,2$, the inclusion $H_{f}(K,T_{i})/p_{n}H_{f}(K,T_{i})↪H_{1}(K,T_{i}/p_{n}T_{i})$ of the finite Bloch-Kato cohomology into the Galois cohomology is functorial with respect to morphisms as $Z/p_{n}Z[G_{K_{0}}]$-modules from $T_{1}/p_{n}T_{1}$ to $T_{2}/p_{n}T_{2}$. In the appendix we give a related result for $p=2$.

## Cite this article

Adrian Iovita, Adriano Marmora, On the continuity of the finite Bloch–Kato cohomology. Rend. Sem. Mat. Univ. Padova 134 (2015), pp. 239–271

DOI 10.4171/RSMUP/134-6