Dieudonné; crystals and Wach modules for p-divisible groups

Let k be a perfect field of characteristic p > 2 and K an extension of F = FracW(k) contained in some F(μ_pr ). Using crystalline Dieudonné; theory, we provide a classification of p-divisible groups over R = O_K[[t₁,. ., t_d]] in terms of finite height (Ρ, τ)-modules over S := W(k)[[u, t₁,. ., t_d]]. When d = 0, such a classification is a consequence of (a special case of) the theory of Kisin-Ren; in this setting, our construction gives an independent proof of this result, and moreover allows us to recover the Dieudonné; crystal of a p-divisible group from the Wach module associated to its Tate module by Berger-Breuil or by Kisin-Ren.