  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.
