On the (, ) -modules corresponding to crystalline representations

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 (, ) -modules over AK^+ and show that it is equivalent to the category of crystalline Zₚ-representations of the absolute Galois group of K. In other words, we determine the (, ) -modules over AK which correspond to crystalline representations. In a sense, this can be seen as a generalization of Wach modules in the ramified case.