On the continuity of the finite Bloch–Kato cohomology

  • Adrian Iovita

    Università di Padova, Italy
  • Adriano Marmora

    Université de Strasbourg, France

Abstract

Let K0K_{0} be an unramified, complete discrete valuation field of mixed characteristics (0,p)(0,p) with perfect residue field. We consider two finite, free Zp{\mathbb{Z}_p}-representations of GK0G_{K_0}, T1T_1 and T2T_2, such that TiZpQpT_i\otimes_{\mathbb{Z}_p} {\mathbb{Q}_p}, for i=1,2i=1,2, are crystalline representations with Hodge-Tate weights between 00 and rp2.r\le p-2. Let KK be a totally ramified extension of degree ee of K0K_0. Supposing that p3p\geq 3 and e(r1)p1e(r-1)\leq p-1, we prove that for every integer n1n\geq 1 and i=1,2i=1,2, the inclusion Hf1(K,Ti)/pnHf1(K,Ti)H1(K,Ti/pnTi)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 Z/pnZ[GK0]\mathbb{Z}/p^n\mathbb{Z}[G_{K_0}]-modules from T1/pnT1T_1/p^nT_1 to T2/pnT2T_2/p^nT_2. In the appendix we give a related result for p=2p=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