On the continuity of the finite Bloch–Kato cohomology

Let K0 be an unramified, complete discrete valuation field of mixed characteristics (0, p) with perfect residue field. We consider two finite, free Zp-representations of GK0 , T1 and T2, such that Ti ⊗Zp Qp, 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 K0. 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 fin(K,Ti)/p H fin(K,Ti) ↪→ H(K,Ti/pTi) of the finite Bloch-Kato cohomology into the Galois cohomology is functorial with respect to morphisms as Z/pZ[GK0 ]-modules from T1/pT1 to T2/pT2. In the appendix we give a related result for p = 2.