Rendiconti del Seminario Matematico della Università di Padova

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Volume 134, 2015, pp. 239–271
DOI: 10.4171/RSMUP/134-6

On the continuity of the finite Bloch–Kato cohomology

Adrian Iovita[1] and Adriano Marmora[2]

(1) Università di Padova, Italy
(2) Université de Strasbourg, France

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 $p\geq 3$ and $e(r-1)\leq p-1$, we prove that for every integer $n\geq 1$ and $i=1,2$, the inclusion $H_f^1(K,T_i)/p^nH_f^1(K,T_i) \hookrightarrow H^1(K, T_i/p^n T_i)$ 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 $T_1/p^nT_1$ to $T_2/p^nT_2$. In the appendix we give a related result for $p=2$.

Keywords: Selmer groups, Galois representations, lattices in crystalline representations, Bloch-Kato finite cohomology, strongly divisible modules

Iovita Adrian, Marmora Adriano: On the continuity of the finite Bloch–Kato cohomology. Rend. Sem. Mat. Univ. Padova 134 (2015), 239-271. doi: 10.4171/RSMUP/134-6