On the $(\varphi,Γ)$-modules corresponding to crystalline representations

Abstract: Let $K$ be a mixed characteristic complete discrete valuation field with perfect residue field of characteristic $p$. We construct a new linear category called the category of crystalline $(\varphi,\Gamma)$-modules over $\widetilde{\mathbb{A}}_K^{+}$ and show that it is equivalent to the category of crystalline $\mathbb{Z}_p$-representations of the absolute Galois group of $K$. In other words, we determine the $(\varphi,\Gamma)$-modules over $\widetilde{\mathbb{A}}_K$ which correspond to crystalline representations. In a sense, this can be seen as a generalization of Wach modules in the ramified case.