ON GALOIS EQUIVARIANCE OF HOMOMORPHISMS BETWEEN TORSION CRYSTALLINE REPRESENTATIONS

Let $K$ be a complete discrete valuation field of mixed characteristic $(0,p)$ with perfect residue field. Let $(\unicode[STIX]{x1D70B}_{n})_{n\geqslant 0}$ be a system of $p$ -power roots of a uniformizer $\unicode[STIX]{x1D70B}=\unicode[STIX]{x1D70B}_{0}$ of $K$ with $\unicode[STIX]{x1D70B}_{n+1}^{p}=\unicode[STIX]{x1D70B}_{n}$ , and define $G_{s}$ (resp. $G_{\infty }$ ) the absolute Galois group of $K(\unicode[STIX]{x1D70B}_{s})$ (resp. $K_{\infty }:=\bigcup _{n\geqslant 0}K(\unicode[STIX]{x1D70B}_{n})$ ). In this paper, we study $G_{s}$ -equivariantness properties of $G_{\infty }$ -equivariant homomorphisms between torsion crystalline representations.