On the $(\phi ,\mathrm{\Gamma })$-modules corresponding to crystalline representations

Let $K$ be a complete discrete valuation field of characteristic $0$ with perfect residue field of characteristic $p>0$. We introduce the notion of crystalline $(\phi ,\mathrm{\Gamma })$-modules over ${\tilde{\mathbb{A}}}_K^{+}$ and show that their category is equivalent to the category of crystalline ${\mathbb{Z}}_p$-representations of the absolute Galois group of $K$. In other words, we determine the $(\phi ,\mathrm{\Gamma })$-modules over ${\tilde{\mathbb{A}}}_K$ that correspond to crystalline representations. This equivalence generalizes, in certain respects, that of L. Berger in the unramified case.