On a theorem of Kisin
Let \(K\) be a complete discrete valuation field of mixed characteristic with perfect residue field. We fix a uniformizer \(\pi\) of \(K\) and consider \(K_\xi = K(\pi^{1/p^\infty})\). The main result of this paper is to give a short and new proof of the following theorem of \textit{M. Kisin} [Prog. Math. 253, 457--496 (2006; Zbl 1184.11052)]: the restriction functor from the category of crystalline representations of \(G_K\) to the category of representations of \(G_{K_\xi}\) is fully faithful. A simple use of Shapiro's lemma reduces the theorem to showing that, if \(R\) is the representation given by inducing the trivial representation of \(G_{K_\xi}\) to \(G_K\), the only non-trivial crystalline \(G_K\)-submodule of \(R\) is just \(\mathbb Q_p\). As \(R\) can be identified with the space of continuous functions on \(G_K/G_{K_\xi}\), the authors prove that the union of all finite-dimensional \(G_K\)-submodules of \(R\) are locally polynomial functions; the union of all semistable \(G_K\)-submodules are polynomial functions; and finally the only nontrivial crystalline \(G_K\)-submodule is the set of constant functions.NEWLINENEWLINEThe paper is very-well written and it only has one page and a half!
scientific article; zbMATH DE number 6216316